Theorem frege65c 38222

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2563 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege59c.a	|- A e. B

Assertion

Ref	Expression
frege65c	|- ( A. x ( ph -> ps ) -> ( A. x ( ps -> ch ) -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) )

Proof of Theorem frege65c

Step	Hyp	Ref	Expression
1		sbcim1 3482	. . 3 |- ( [. A / x ]. ( ph -> ps ) -> ( [. A / x ]. ph -> [. A / x ]. ps ) )
2		frege59c.a	. . . 4 |- A e. B
3	2	frege64c 38221	. . 3 |- ( ( [. A / x ]. ph -> [. A / x ]. ps ) -> ( A. x ( ps -> ch ) -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) )
4	1, 3	syl 17	. 2 |- ( [. A / x ]. ( ph -> ps ) -> ( A. x ( ps -> ch ) -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) )
5	2	frege61c 38218	. 2 |- ( ( [. 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 5	1 |- ( A. x ( ph -> ps ) -> ( A. x ( ps -> ch ) -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) )

Syntax hints:   -> wi 4   A.wal 1481   e. wcel 1990   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-12 2047  ax-13 2246  ax-ext 2602  ax-frege1 38084  ax-frege2 38085  ax-frege8 38103  ax-frege58b 38195

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  frege66c  38223