So i have to prove that: $$\{\neg A\to B,A\to C,B\to D\}\vdash \neg C\to D$$
I can use logical axioms, modus ponens and 'metatheorems'.
Logical axioms:
- φ→(ψ→φ)
- (φ→(ψ→χ))→((φ→ψ)→(φ→χ))
- (¬φ→¬ψ)→(ψ→φ)
Also i can use modus ponens(the only rule i can use) and metatheorems
Some thoughts:So i started experimenting with all $3$ tools i have, started asking myself is any of the hypotheses can give as something new using the logic axioms but then i stalled, and modus ponens can't do much on it's own knowing these hypotheses atleast.My next thought was that i have to use those 2 metatheorems in order to actualy prove one part of $\neg C\to D$ (based on metatheorem 2) meaning i use as a hypothesis $\neg C$ to prove $D$ but i am stuck and i don't undestand even how to start.
Best Answer
Hint
1) Using Modus Ponens and your Metatheorem 1, prove Hypothetical Syllogism :
2) Using axioms, prove Contraposition :
Finally, use them to derive :