Metamath Proof Explorer

Theorem frege65c

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara when the minor premise has a general context. Proposition 65 of Frege1879 p. 53. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

Ref Expression
Hypothesis frege59c.a 
|- A e. B
Assertion frege65c 
|- ( A. x ( ph -> ps ) -> ( A. x ( ps -> ch ) -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) )

Proof

Step Hyp Ref Expression
1 frege59c.a 
 |-  A e. B
2 sbcim1 
 |-  ( [. A / x ]. ( ph -> ps ) -> ( [. A / x ]. ph -> [. A / x ]. ps ) )
3 1 frege64c 
 |-  ( ( [. A / x ]. ph -> [. A / x ]. ps ) -> ( A. x ( ps -> ch ) -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) )
4 2 3 syl 
 |-  ( [. A / x ]. ( ph -> ps ) -> ( A. x ( ps -> ch ) -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) )
5 1 frege61c 
 |-  ( ( [. A / x ]. ( ph -> ps ) -> ( A. x ( ps -> ch ) -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) ) -> ( A. x ( ph -> ps ) -> ( A. x ( ps -> ch ) -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) ) )
6 4 5 ax-mp 
 |-  ( A. x ( ph -> ps ) -> ( A. x ( ps -> ch ) -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) )