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My question is how should I use propositional logic rules for → and ↔ (although other rules may be required) to prove the following:

  1. A → B, B → C ⊢ (AvB) → C

  2. A ↔ B ⊢ ¬A ↔ ¬B

Please use the language of propositional logic.

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Here's an answer to your first question using conditional proof. If you're unfamiliar with it, conditional proof is a proof method in PL that allows you to assume the antecedent of a conclusion (assuming, of course, that the main operator of your conclusion is a conditional), and allows you to derive a conditional whose consequent is whatever you can deduce within the scope of your assumption. This is what's going on on lines 4-6. Notice the indentation - by convention, we indent whatever follows from our assumption, only returning to the non-indented position when we discharge our assumption by deriving the conditional (again, a conditional whose antecedent == whatever you've assumed).

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