# fitch proof chapter 13 exercise 13.49 [closed]

Does anyone know how to solve 13.49

``````∃x P(x)
∀x ∀y ((P(x) ∧ P(y)) → x = y)

= ∃x (P(x) ∧ ∀y (P(y) → y = x))
``````

and 13.50

∃x (P(x) ∧ ∀y (P(y) → y = x))

= ∀x ∀y ((P(x) ∧ P(y)) → x = y)

I have big problems!

## closed as off-topic by Mauro ALLEGRANZA, Keelan♦Apr 20 '17 at 7:47

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context. Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Keelan
If this question can be reworded to fit the rules in the help center, please edit the question.

• First problem. Ask one question per question... Second problem, the formatting (I can fix the second problem) – virmaior Apr 19 '17 at 11:15
• Quite similar to this post – Mauro ALLEGRANZA Apr 19 '17 at 12:16
• Also, when referring to exercise numbers, you should mention the book you're taking them from. – Keelan Apr 19 '17 at 19:45

```{1}      1.  Ǝx[Px]                            Prem.
{2}      2.  ∀x∀y[(Px & Py) → x=y]            Prem.
{3}      3.  Pa                                Assum.
{2}      4.  ∀y[(Pa & Py) → a=y]               2 UE
{5}      5.  Pb                                Assum. TD(b)
{3,5}    6.  Pa & Pb                           3,5 &I
{2}      7.  (Pa & Pb) → a=b                   4 UE
{2,3,5}  8.  a=b                               6,7 MP
{2,5}    9.  Pa → a=b                          3,8 CI
{2,5}    10. ∀y[Py → y=b]                     9 UI
{2,5}    11. Pb & ∀y[Py → y=b]                5,10 &I
{2,5}    12. Ǝx[Px & ∀y[Py → y=x]]            11 EI
{1,2}    13. Ǝx[Px & ∀y[Py → y=x]]            1,5,12 EE
```

Here's the second one:

```{1}      1.  Ǝx[Px & ∀y[Py → y=x]]            Prem.
{2}      2.  Pa & Pb                          Assum.
{3}      3.  Pc & ∀y[Py → y=c]                Assum. TD(c)
{3}      4.  ∀y[Py → y=c]                     3 &E
{3}      5.  Pa → a=c                         4 UE
{2}      6.  Pa                               2 &E
{2,3}    7.  a=c                              5,6 MP
{3}      8.  Pb → b=c                         4 UE
{2}      9.  Pb                               2 &E
{2,3}    10. b=c                              8,9 MP
{2,3}    11. a=b                              7,10 =E
{3}      12. (Pa & Pb) → a=b                  2,11 MP
{3}      13. ∀y[(Pa & Py) → a=y]              12 UI
{3}      14. ∀x∀y[(Px & Py) → x=y]            13 UI
{1}      i5. ∀x∀y[(Px & Py) → x=y]            1,3,14 EE
```