Theorem frege62c

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara when the minor premise has a particular context. Proposition 62 of Frege1879 p. 52. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege59c.a	|- A e. B
Assertion	frege62c	|- ( [. A / x ]. ph -> ( A. x ( ph -> ps ) -> [. A / x ]. ps ) )

Proof

Step	Hyp	Ref	Expression
1		frege59c.a	|- A e. B
2	1	frege58c	|- ( A. x ( ph -> ps ) -> [. A / x ]. ( ph -> ps ) )
3		sbcim1	|- ( [. A / x ]. ( ph -> ps ) -> ( [. A / x ]. ph -> [. A / x ]. ps ) )
4	2 3	syl	|- ( A. x ( ph -> ps ) -> ( [. A / x ]. ph -> [. A / x ]. ps ) )
5		ax-frege8	|- ( ( A. x ( ph -> ps ) -> ( [. A / x ]. ph -> [. A / x ]. ps ) ) -> ( [. A / x ]. ph -> ( A. x ( ph -> ps ) -> [. A / x ]. ps ) ) )
6	4 5	ax-mp	|- ( [. A / x ]. ph -> ( A. x ( ph -> ps ) -> [. A / x ]. ps ) )