Metamath Proof Explorer

Theorem frege63c

Description: Analogue of frege63b . Proposition 63 of Frege1879 p. 52. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

Ref Expression
Hypothesis frege59c.a 
|- A e. B
Assertion frege63c 
|- ( [. A / x ]. ph -> ( ps -> ( A. x ( ph -> ch ) -> [. A / x ]. ch ) ) )

Proof

Step Hyp Ref Expression
1 frege59c.a 
 |-  A e. B
2 1 frege62c 
 |-  ( [. A / x ]. ph -> ( A. x ( ph -> ch ) -> [. A / x ]. ch ) )
3 frege24 
 |-  ( ( [. A / x ]. ph -> ( A. x ( ph -> ch ) -> [. A / x ]. ch ) ) -> ( [. A / x ]. ph -> ( ps -> ( A. x ( ph -> ch ) -> [. A / x ]. ch ) ) ) )
4 2 3 ax-mp 
 |-  ( [. A / x ]. ph -> ( ps -> ( A. x ( ph -> ch ) -> [. A / x ]. ch ) ) )