Theorem frege54cor1c 38209

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 3213	. . . 4 |- A e. _V
3	2	snid 4208	. . 3 |- A e. { A }
4		df-sn 4178	. . 3 |- { A } = { x | x = A }
5	3, 4	eleqtri 2699	. 2 |- A e. { x | x = A }
6		df-sbc 3436	. 2 |- ( [. A / x ]. x = A <-> A e. { x | x = A } )
7	5, 6	mpbir 221	1 |- [. A / x ]. x = A

Syntax hints:   = wceq 1483   e. wcel 1990   {cab 2608   [.wsbc 3435   {csn 4177

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  df-sn 4178

This theorem is referenced by:  frege55lem2c  38211  frege55c  38212  frege56c  38213