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	|- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) )

Proof

Step	Hyp	Ref	Expression
1		ax-frege1	|- ( ps -> ( ph -> ps ) )
2		frege9	|- ( ( ps -> ( ph -> ps ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) ) )
3	1 2	ax-mp	|- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) )