Theorem frege11 38108

Description: Elimination of a nested antecedent as a partial converse of ja 173. 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 106. Proposition 11 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege11	|- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) )

Proof of Theorem frege11

Step	Hyp	Ref	Expression
1		ax-frege1 38084	. 2 |- ( ps -> ( ph -> ps ) )
2		frege9 38106	. 2 |- ( ( ps -> ( ph -> ps ) ) -> ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) ) )
3	1, 2	ax-mp 5	1 |- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) )

Syntax hints:   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 38084  ax-frege2 38085  ax-frege8 38103

This theorem is referenced by:  frege112  38269