Theorem frege68c 38225

Description: Combination of applying a definition and applying it to a specific instance. Proposition 68 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
frege68c	|- ( ( A. x ph <-> ps ) -> ( ps -> [. A / x ]. ph ) )

Proof of Theorem frege68c

Step	Hyp	Ref	Expression
1		frege57aid 38166	. 2 |- ( ( A. x ph <-> ps ) -> ( ps -> A. x ph ) )
2		frege59c.a	. . 3 |- A e. B
3	2	frege67c 38224	. 2 |- ( ( ( A. x ph <-> ps ) -> ( ps -> A. x ph ) ) -> ( ( A. x ph <-> ps ) -> ( ps -> [. A / x ]. ph ) ) )
4	1, 3	ax-mp 5	1 |- ( ( 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-frege8 38103  ax-frege52a 38151  ax-frege58b 38195

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-tru 1486  df-fal 1489  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:  frege70  38227  frege77  38234  frege116  38273