Theorem frege59c

Description: A kind of Aristotelian inference. Proposition 59 of Frege1879 p. 51.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collectionFrom Frege to Goedel, this proof has the frege12 incorrectly referenced where frege30 is in the original. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege59c.a	|- A e. B
Assertion	frege59c	|- ( [. A / x ]. ph -> ( -. [. A / x ]. ps -> -. A. x ( ph -> 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		frege30	|- ( ( A. x ( ph -> ps ) -> ( [. A / x ]. ph -> [. A / x ]. ps ) ) -> ( [. A / x ]. ph -> ( -. [. A / x ]. ps -> -. A. x ( ph -> ps ) ) ) )
6	4 5	ax-mp	|- ( [. A / x ]. ph -> ( -. [. A / x ]. ps -> -. A. x ( ph -> ps ) ) )