Theorem frege57c 38214

Description: Swap order of implication in ax-frege52c 38182. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege57c.a	|- A e. C

Assertion

Ref	Expression
frege57c	|- ( A = B -> ( [. B / x ]. ph -> [. A / x ]. ph ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   C( x)

Proof of Theorem frege57c

Step	Hyp	Ref	Expression
1		ax-frege52c 38182	. 2 |- ( B = A -> ( [. B / x ]. ph -> [. A / x ]. ph ) )
2		frege57c.a	. . 3 |- A e. C
3	2	frege56c 38213	. 2 |- ( ( B = A -> ( [. B / x ]. ph -> [. A / x ]. ph ) ) -> ( A = B -> ( [. B / x ]. ph -> [. A / x ]. ph ) ) )
4	1, 3	ax-mp 5	1 |- ( A = B -> ( [. B / x ]. ph -> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   = wceq 1483   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-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-frege1 38084  ax-frege2 38085  ax-frege8 38103  ax-frege52c 38182

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-nfc 2753  df-v 3202  df-sbc 3436  df-sn 4178

This theorem is referenced by: (None)