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{1}      1.   ~A ∨ (A → B)                Prem.
{2}      2.   A                           Assum.
{3}      3.   ~A                          Assum. (1st D)
{4}      4.   ~B                          Assum.
{3,4}    5.   ~A & ~B                     3,4 &I
{3,4}    6.   ~A                          5 &E
{2,3,4}  7.   A & ~A                      2,6 &I
{2,3}    8.   B                           4,7 RAA (1st C)                       
{9}      9.   A → B                       Assum. (2nd D)
{2,9}    10.  B                           2,9 MP (2nd C)
{1,2}    11.  B                           1,3,8,9,10 ∨E
{1}      12.  A → B                       2,3 CP    

The best way to prove an implication is usually to start by assuming the antecedent of the implication to be concluded (line 2). To eliminate the disjunction, I also have to assume each of the disjuncts (lines 3 and 9). The first disjunct contradicts the assumption in line 2, and given a contradiction, you can essentially conclude whatever you want. I needed to conclude B, but it should be noted that I didn't get that conclusion for free; it depends on the assumptions of lines 2 and 3 which will have to be eliminated (line 8). The second disjunct also lets me conclude B using modus ponens. Since both disjuncts imply the same conclusion of B (line 11), I can eliminate the assumptions of lines 3 and 9. The assumption on line 2 led to the conclusion on line 11, so I can eliminate the initial assumption and conclude that A → B.

{1}      1.   ~A ∨ (A → B)                Prem.
{2}      2.   A                           Assum.
{3}      3.   ~A                          Assum. (1st D)
{4}      4.   ~B                          Assum.
{3,4}    5.   ~A & ~B                     3,4 &I
{3,4}    6.   ~A                          5 &E
{2,3,4}  7.   A & ~A                      2,6 &I
{2,3}    8.   B                           4,7 RAA (1st C)                       
{9}      9.   A → B                       Assum. (2nd D)
{2,9}    10.  B                           2,9 MP (2nd C)
{1,2}    11.  B                           1,3,8,9,10 ∨E
{1}      12.  A → B                       2,11 CP    

The best way to prove an implication is usually to start by assuming the antecedent of the implication to be concluded (line 2). To eliminate the disjunction, I also have to assume each of the disjuncts (lines 3 and 9). The first disjunct contradicts the assumption in line 2, and given a contradiction, you can essentially conclude whatever you want. I needed to conclude B, but it should be noted that I didn't get that conclusion for free; it depends on the assumptions of lines 2 and 3 which will have to be eliminated (line 8). The second disjunct also lets me conclude B using modus ponens. Since both disjuncts imply the same conclusion of B (line 11), I can eliminate the assumptions of lines 3 and 9. The assumption on line 2 led to the conclusion on line 11, so I can eliminate it and conclude that A → B.

Edit:

Due to a question in a comment, I'm adding the following proof:

{1}      1.   ~A ∨ (A → B)                Prem.
{2}      2.   A                           Assum.
{3}      3.   ~A                          Assum. (1st D)
{4}      4.   ~B                          Assum.
{3,4}    5.   ~A & ~B                     3,4 &I
{3,4}    6.   ~A                          5 &E
{2,3,4}  7.   A & ~A                      2,6 &I
{2,3}    8.   B                           4,7 RAA (1st C)                       
{9}      9.   A → B                       Assum. (2nd D)
{2,9}    10.  B                           2,9 MP (2nd C)
{1,2}    11.  B                           1,3,8,9,10 ∨E
{1}      12.  A → B                       2,3 CP    

The best way to prove an implication is usually to start by assuming the antecedent of the implication to be concluded (line 2). To eliminate the disjunction, I also have to assume each of the disjuncts (lines 3 and 9). The first disjunct contradicts the assumption in line 2, and given a contradiction, you can essentially conclude whatever you want. I needed to conclude B, but it should be noted that I didn't get that conclusion for free; it depends on the assumptions of lines 2 and 3 which will have to be eliminated (line 8). The second disjunct also lets me conclude B using modus ponens. Since both disjuncts imply the same conclusion of B (line 11), I can eliminate the assumptions of lines 3 and 9. The assumption on line 2 led to the conclusion on line 11, so I can eliminate the initial assumption and conclude that A → B.

It might be helpful to think of this by considering each of the disjuncts of a disjunctive statement as containing unspoken implications. Consider, for example, the following disjunction:

Either no storm comes, or the boat will sink.

For each of the disjuncts, we can assert the following implications:

• If no storm comes, no storm comes.
• If a storm comes, the boat will sink.

It's intuitively true that either a storm will come or not. Therefore, we can put all that together to derive the following, which is a logical consequence to the first statement:

If no storm comes, no storm comes; or if a storm does come, the boat will sink.

The first part is redundant, so this can be simplified to:

Either no storm comes; or if a storm comes, the boat will sink.

From this it can be seen that disjunctive statements can be thought of as having conditional statements hidden within their elements.

Vacuity

It might be argued that it is vacuous to insert an antecedent into the disjunction in that way. For example, it could be argued that any antecedent could be inserted, such as:

Either no storm comes; or if the moon is made of cheese, the boat will sink.

Although it is true that that logically follows from the premise, there's an important difference to be noted. The antecedent "if a storm comes" complements the other disjunct, whereas "if the moon is made of cheese" does not. This involves the important difference that the simpler conditional follows logically from the disjunctive statement:

• ~A ∨ (A → B) implies A → B
• But, ~A ∨ (C → B) does not imply C → B

In other words, it cannot be concluded that if the moon is made of cheese, the boat will sink; but, it can be concluded that if a storm does come, the boat will sink.

It might be helpful to think of this by considering each of the disjuncts of a disjunctive statement as containing unspoken implications. Consider, for example, the following disjunction:

Either no storm will come, or the boat will sink.

For each of the disjuncts, we can assert the following implications:

• If no storm comes, no storm comes.
• If a storm comes, the boat will sink.

It's intuitively true that either a storm will come or not. Therefore, we can put all that together to derive the following, which is a logical consequence to the first statement:

If no storm comes, no storm comes; or if a storm does come, the boat will sink.

The first part is redundant, so this can be simplified to:

Either no storm comes; or if a storm comes, the boat will sink.

From this it can be seen that disjunctive statements can be thought of as having conditional statements hidden within their elements.

Vacuity

It might be argued that it is vacuous to insert an antecedent into the disjunction in that way. For example, it could be argued that any antecedent could be inserted, such as:

Either no storm comes; or if the moon is made of cheese, the boat will sink.

Although it is true that that logically follows from the premise, there's an important difference to be noted. The antecedent "if a storm comes" complements the other disjunct, whereas "if the moon is made of cheese" does not. This involves the important difference that the simpler conditional follows logically from the disjunctive statement:

• ~A ∨ (A → B) implies A → B
• But, ~A ∨ (C → B) does not imply C → B

In other words, it cannot be concluded that if the moon is made of cheese, the boat will sink; but, it can be concluded that if a storm does come, the boat will sink.