Theorem frege29

Description: Closed form of con3d . Proposition 29 of Frege1879 p. 43. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege29	$$⊢\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\phi \to \left(\neg \chi \to \neg \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-frege28	$$⊢\left(\psi \to \chi \right)\to \left(\neg \chi \to \neg \psi \right)$$
2		frege5	$$⊢\left(\left(\psi \to \chi \right)\to \left(\neg \chi \to \neg \psi \right)\right)\to \left(\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\phi \to \left(\neg \chi \to \neg \psi \right)\right)\right)$$
3	1 2	ax-mp	$$⊢\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\phi \to \left(\neg \chi \to \neg \psi \right)\right)$$