Theorem frege7 38102

Description: A closed form of syl6 35. The first antecedent is used to replace the consequent of the second antecedent. Proposition 7 of [Frege1879] p. 34. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege7	|- ( ( ph -> ps ) -> ( ( ch -> ( th -> ph ) ) -> ( ch -> ( th -> ps ) ) ) )

Proof of Theorem frege7

Step	Hyp	Ref	Expression
1		frege5 38094	. 2 |- ( ( ph -> ps ) -> ( ( th -> ph ) -> ( th -> ps ) ) )
2		frege6 38100	. 2 |- ( ( ( ph -> ps ) -> ( ( th -> ph ) -> ( th -> ps ) ) ) -> ( ( ph -> ps ) -> ( ( ch -> ( th -> ph ) ) -> ( ch -> ( th -> ps ) ) ) ) )
3	1, 2	ax-mp 5	1 |- ( ( ph -> ps ) -> ( ( ch -> ( th -> ph ) ) -> ( ch -> ( th -> ps ) ) ) )

Syntax hints:   -> wi 4

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

This theorem is referenced by:  frege32  38129  frege67a  38179  frege67b  38206  frege67c  38224  frege94  38251  frege107  38264  frege113  38270