Metamath Proof Explorer

Theorem frege60c

Description: Swap antecedents of frege58c . Proposition 60 of Frege1879 p. 52. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

Ref Expression
Hypothesis frege59c.a 
|- A e. B
Assertion frege60c 
|- ( A. x ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ps -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) )

Proof

Step Hyp Ref Expression
1 frege59c.a 
 |-  A e. B
2 1 frege58c 
 |-  ( A. x ( ph -> ( ps -> ch ) ) -> [. A / x ]. ( ph -> ( ps -> ch ) ) )
3 sbcim1 
 |-  ( [. A / x ]. ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) )
4 sbcim1 
 |-  ( [. A / x ]. ( ps -> ch ) -> ( [. A / x ]. ps -> [. A / x ]. ch ) )
5 3 4 syl6 
 |-  ( [. A / x ]. ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) )
6 2 5 syl 
 |-  ( A. x ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) )
7 frege12 
 |-  ( ( A. x ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) -> ( A. x ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ps -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) ) )
8 6 7 ax-mp 
 |-  ( A. x ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ps -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) )