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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
StepHypRef Expression
1 frege54c.1 . . . . 5  |-  A  e.  C
21elexi 3041 . . . 4  |-  A  e. 
_V
32snid 3988 . . 3  |-  A  e. 
{ A }
4 df-sn 3960 . . 3  |-  { A }  =  { x  |  x  =  A }
53, 4eleqtri 2547 . 2  |-  A  e. 
{ x  |  x  =  A }
6 df-sbc 3256 . 2  |-  ( [. A  /  x ]. x  =  A  <->  A  e.  { x  |  x  =  A } )
75, 6mpbir 214 1  |-  [. A  /  x ]. x  =  A
Colors of variables: wff setvar class
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
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