# How to deduce V from (U → V ), (¬U→V) using Deduction theorem?

We have these axioms and modus ponens:

Axiom 1 is: P→(Q→P)

Axiom 2 is: (P→(Q→R))→((P→Q)→(P→R))

Axiom 3 is:(P→Q)→(¬Q→¬P)

Modus ponens is: from P and P→Q infer Q

Edit: I still couldn't solve it, so I would really appreciate it if someone gave me a hint

Using these axioms and modus penons I want to infer V from (U → V ), (¬U→V)

Using axiom 3 I said that (V→ (U → V ) ) → (¬ (U →V ) → ¬V ). And using axiom 1 I said that V→ (U → V ). So using modus ponens we can infer that ¬ (U →V ) → ¬V.

If I could somehow write ¬ (U →V ) using one of these axioms then I could infer ¬V.

I'm a bit stuck here. Any help please?

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• I’m voting to close this question because this is not a homework forum – Swami Vishwananda Nov 6 '20 at 4:25
• It's not homework – aneys Nov 7 '20 at 10:57