I am trying to prove that the following is a tautology:
$(A \implies (B \implies C)) \implies ((A \implies (C \implies D)) \implies (A \implies (B \implies D)))$
To make progress, I thought I'd eliminate all the arrows. After that, and some de Morgan, I've arrived at:
$(A \land B \land ¬C) \lor (A \land ¬C \land ¬D) \lor (¬A \lor B \lor D) $
At this point, I don't know how to carry on, though. I feel like I'm missing some rule -- I get stuck in trying to expand this and don't get anywhere.
I'd be really grateful for help / hints!
Thank you Henning Makholm and Mauro ALLEGRANZA for spotting mistakes in my reformulations. The rewritten form should read:
$(A \land B \land ¬C) \lor (A \land C \land ¬D) \lor (¬A \lor ¬B \lor D) $