Metamath Proof Explorer

Theorem frege67c

Description: Lemma for frege68c . Proposition 67 of Frege1879 p. 54. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

Ref Expression
Hypothesis frege59c.a 
|- A e. B
Assertion frege67c 
|- ( ( ( A. x ph <-> ps ) -> ( ps -> A. x ph ) ) -> ( ( A. x ph <-> ps ) -> ( ps -> [. A / x ]. ph ) ) )

Proof

Step Hyp Ref Expression
1 frege59c.a 
 |-  A e. B
2 1 frege58c 
 |-  ( A. x ph -> [. A / x ]. ph )
3 frege7 
 |-  ( ( A. x ph -> [. A / x ]. ph ) -> ( ( ( A. x ph <-> ps ) -> ( ps -> A. x ph ) ) -> ( ( A. x ph <-> ps ) -> ( ps -> [. A / x ]. ph ) ) ) )
4 2 3 ax-mp 
 |-  ( ( ( A. x ph <-> ps ) -> ( ps -> A. x ph ) ) -> ( ( A. x ph <-> ps ) -> ( ps -> [. A / x ]. ph ) ) )