Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  elabgf0 Unicode version

Theorem elabgf0 15214
Description: Lemma for elabgf 2902. (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 2256 . 2  |-  ( x  =  A  ->  (
x  e.  { x  |  ph }  <->  A  e.  { x  |  ph }
) )
2 abid 2181 . 2  |-  ( x  e.  { x  | 
ph }  <->  ph )
31, 2bitr3di 195 1  |-  ( x  =  A  ->  ( A  e.  { x  |  ph }  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1364    e. wcel 2164   {cab 2179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189
This theorem is referenced by:  elabgft1  15215  elabgf2  15217
  Copyright terms: Public domain W3C validator