Theorem frege55lem1c 36557

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 3280	. . 3 |- ( [. A / x ]. x = B <-> A e. { x | x = B } )
2		eqeq1 2466	. . . . 5 |- ( x = A -> ( x = B <-> A = B ) )
3	2	elabg 3198	. . . 4 |- ( A e. { x | x = B } -> ( A e. { x | x = B } <-> A = B ) )
4	3	ibi 249	. . 3 |- ( A e. { x | x = B } -> A = B )
5	1, 4	sylbi 200	. 2 |- ( [. A / x ]. x = B -> A = B )
6	5	imim2i 16	1 |- ( ( ph -> [. A / x ]. x = B ) -> ( ph -> A = B ) )

Syntax hints:   -> wi 4   = wceq 1455   e. wcel 1898   {cab 2448   [.wsbc 3279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-sbc 3280

This theorem is referenced by:  frege56c  36560