Theorem frege55lem1c 38133

Description: Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)

Assertion

Ref	Expression
frege55lem1c	|- ( ( ph -> [. A / x ]. x = B ) -> ( ph -> A = B ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem frege55lem1c

Step	Hyp	Ref	Expression
1		df-sbc 3328	. . 3 |- ( [. A / x ]. x = B <-> A e. { x | x = B } )
2		ax-frege1 38003	. . . . 5 |- ( A e. { x | x = B } -> ( A = B -> A e. { x | x = B } ) )
3		eqeq1 2461	. . . . . 6 |- ( x = A -> ( x = B <-> A = B ) )
4	3	elab3g 3252	. . . . 5 |- ( ( A = B -> A e. { x | x = B } ) -> ( A e. { x | x = B } <-> A = B ) )
5	2, 4	syl 16	. . . 4 |- ( A e. { x | x = B } -> ( A e. { x | x = B } <-> A = B ) )
6	5	ibi 241	. . 3 |- ( A e. { x | x = B } -> A = B )
7	1, 6	sylbi 195	. 2 |- ( [. A / x ]. x = B -> A = B )
8	7	imim2i 14	1 |- ( ( ph -> [. A / x ]. x = B ) -> ( ph -> A = B ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1395   e. wcel 1819   {cab 2442   [.wsbc 3327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-frege1 38003

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328

This theorem is referenced by:  frege56c  38136