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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
StepHypRef Expression
1 frege54c.1 . . . . 5  |-  A  e.  C
21elexi 3055 . . . 4  |-  A  e. 
_V
32snid 3996 . . 3  |-  A  e. 
{ A }
4 df-sn 3969 . . 3  |-  { A }  =  { x  |  x  =  A }
53, 4eleqtri 2527 . 2  |-  A  e. 
{ x  |  x  =  A }
6 df-sbc 3268 . 2  |-  ( [. A  /  x ]. x  =  A  <->  A  e.  { x  |  x  =  A } )
75, 6mpbir 213 1  |-  [. A  /  x ]. x  =  A
Colors of variables: wff setvar class
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
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