Theorem frege56c 38213

Description: Lemma for frege57c 38214. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege56c.b	|- B e. C

Assertion

Ref	Expression
frege56c	|- ( ( A = B -> ( [. A / x ]. ph -> [. B / x ]. ph ) ) -> ( B = A -> ( [. A / x ]. ph -> [. B / x ]. ph ) ) )

Distinct variable groups:   x, A   x, B

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

Proof of Theorem frege56c

Step	Hyp	Ref	Expression
1		frege56c.b	. . . . 5 |- B e. C
2	1	frege54cor1c 38209	. . . 4 |- [. B / x ]. x = B
3		frege53c 38208	. . . 4 |- ( [. B / x ]. x = B -> ( B = A -> [. A / x ]. x = B ) )
4	2, 3	ax-mp 5	. . 3 |- ( B = A -> [. A / x ]. x = B )
5		frege55lem1c 38210	. . 3 |- ( ( B = A -> [. A / x ]. x = B ) -> ( B = A -> A = B ) )
6	4, 5	ax-mp 5	. 2 |- ( B = A -> A = B )
7		frege9 38106	. 2 |- ( ( B = A -> A = B ) -> ( ( A = B -> ( [. A / x ]. ph -> [. B / x ]. ph ) ) -> ( B = A -> ( [. A / x ]. ph -> [. B / x ]. ph ) ) ) )
8	6, 7	ax-mp 5	1 |- ( ( A = B -> ( [. A / x ]. ph -> [. B / x ]. ph ) ) -> ( B = A -> ( [. A / x ]. ph -> [. B / 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:  frege57c  38214