Theorem frege55lem1c 38210

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 3436	. . 3 |- ( [. A / x ]. x = B <-> A e. { x | x = B } )
2		eqeq1 2626	. . . . 5 |- ( x = A -> ( x = B <-> A = B ) )
3	2	elabg 3351	. . . 4 |- ( A e. { x | x = B } -> ( A e. { x | x = B } <-> A = B ) )
4	3	ibi 256	. . 3 |- ( A e. { x | x = B } -> A = B )
5	1, 4	sylbi 207	. 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 1483   e. wcel 1990   {cab 2608   [.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

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

This theorem is referenced by:  frege56c  38213