Theorem frege32

Description: Deduce con1 from con3 . Proposition 32 of Frege1879 p. 44. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege32	$$⊢\left(\left(\neg \phi \to \psi \right)\to \left(\neg \psi \to \neg \neg \phi \right)\right)\to \left(\left(\neg \phi \to \psi \right)\to \left(\neg \psi \to \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-frege31	$$⊢\neg \neg \phi \to \phi$$
2		frege7	$$⊢\left(\neg \neg \phi \to \phi \right)\to \left(\left(\left(\neg \phi \to \psi \right)\to \left(\neg \psi \to \neg \neg \phi \right)\right)\to \left(\left(\neg \phi \to \psi \right)\to \left(\neg \psi \to \phi \right)\right)\right)$$
3	1 2	ax-mp	$$⊢\left(\left(\neg \phi \to \psi \right)\to \left(\neg \psi \to \neg \neg \phi \right)\right)\to \left(\left(\neg \phi \to \psi \right)\to \left(\neg \psi \to \phi \right)\right)$$