Theorem frege54cor1c 36582

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 3041	. . . 4 |- A e. _V
3	2	snid 3988	. . 3 |- A e. { A }
4		df-sn 3960	. . 3 |- { A } = { x | x = A }
5	3, 4	eleqtri 2547	. 2 |- A e. { x | x = A }
6		df-sbc 3256	. 2 |- ( [. A / x ]. x = A <-> A e. { x | x = A } )
7	5, 6	mpbir 214	1 |- [. A / x ]. x = A

Syntax hints:   = wceq 1452   e. wcel 1904   {cab 2457   [.wsbc 3255   {csn 3959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-sbc 3256  df-sn 3960

This theorem is referenced by:  frege55lem2c  36584  frege55c  36585  frege56c  36586