Theorem frege54cor1c 36511

Description: Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Revised by RP, 25-Apr-2020.)

Hypothesis

Ref	Expression
frege54c.1	|- A e. C

Assertion

Ref	Expression
frege54cor1c	|- [. A / x ]. x = A

Distinct variable group:   x, A

Allowed substitution hint:   C( x)

Proof of Theorem frege54cor1c

Step	Hyp	Ref	Expression
1		frege54c.1	. . . . 5 |- A e. C
2	1	elexi 3055	. . . 4 |- A e. _V
3	2	snid 3996	. . 3 |- A e. { A }
4		df-sn 3969	. . 3 |- { A } = { x | x = A }
5	3, 4	eleqtri 2527	. 2 |- A e. { x | x = A }
6		df-sbc 3268	. 2 |- ( [. A / x ]. x = A <-> A e. { x | x = A } )
7	5, 6	mpbir 213	1 |- [. A / x ]. x = A

Syntax hints:   = wceq 1444   e. wcel 1887   {cab 2437   [.wsbc 3267   {csn 3968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-sbc 3268  df-sn 3969

This theorem is referenced by:  frege55lem2c  36513  frege55c  36514  frege56c  36515