Theorem frege11

Description: Elimination of a nested antecedent as a partial converse of ja . If the proposition that ps takes place or ph does not is a sufficient condition for ch , then ps by itself is a sufficient condition for ch . Identical to jarr . Proposition 11 of Frege1879 p. 36. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege11	$$⊢\left(\left(\phi \to \psi \right)\to \chi \right)\to \left(\psi \to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-frege1	$$⊢\psi \to \left(\phi \to \psi \right)$$
2		frege9	$$⊢\left(\psi \to \left(\phi \to \psi \right)\right)\to \left(\left(\left(\phi \to \psi \right)\to \chi \right)\to \left(\psi \to \chi \right)\right)$$
3	1 2	ax-mp	$$⊢\left(\left(\phi \to \psi \right)\to \chi \right)\to \left(\psi \to \chi \right)$$