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Theorem elabgf0 13014
Description: Lemma for elabgf 2826. (Contributed by BJ, 21-Nov-2019.)
Assertion
Ref Expression
elabgf0  |-  ( x  =  A  ->  ( A  e.  { x  |  ph }  <->  ph ) )

Proof of Theorem elabgf0
StepHypRef Expression
1 eleq1 2202 . 2  |-  ( x  =  A  ->  (
x  e.  { x  |  ph }  <->  A  e.  { x  |  ph }
) )
2 abid 2127 . 2  |-  ( x  e.  { x  | 
ph }  <->  ph )
31, 2bitr3di 194 1  |-  ( x  =  A  ->  ( A  e.  { x  |  ph }  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    e. wcel 1480   {cab 2125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135
This theorem is referenced by:  elabgft1  13015  elabgf2  13017
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