Theorem frege67c 38224

Description: Lemma for frege68c 38225. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege59c.a	|- A e. B

Assertion

Ref	Expression
frege67c	|- ( ( ( A. x ph <-> ps ) -> ( ps -> A. x ph ) ) -> ( ( A. x ph <-> ps ) -> ( ps -> [. A / x ]. ph ) ) )

Proof of Theorem frege67c

Step	Hyp	Ref	Expression
1		frege59c.a	. . 3 |- A e. B
2	1	frege58c 38215	. 2 |- ( A. x ph -> [. A / x ]. ph )
3		frege7 38102	. 2 |- ( ( A. x ph -> [. A / x ]. ph ) -> ( ( ( A. x ph <-> ps ) -> ( ps -> A. x ph ) ) -> ( ( A. x ph <-> ps ) -> ( ps -> [. A / x ]. ph ) ) ) )
4	2, 3	ax-mp 5	1 |- ( ( ( A. x ph <-> ps ) -> ( ps -> A. x ph ) ) -> ( ( A. x ph <-> ps ) -> ( ps -> [. A / x ]. ph ) ) )

Syntax hints:   -> wi 4   <-> wb 196   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-12 2047  ax-ext 2602  ax-frege1 38084  ax-frege2 38085  ax-frege58b 38195

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

This theorem is referenced by:  frege68c  38225