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Theorem elabg2 9924
Description: One implication of elabg 2688. (Contributed by BJ, 21-Nov-2019.)
Hypothesis
Ref Expression
elabg2.1  |-  ( x  =  A  ->  ( ps  ->  ph ) )
Assertion
Ref Expression
elabg2  |-  ( A  e.  V  ->  ( ps  ->  A  e.  {
x  |  ph }
) )
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem elabg2
StepHypRef Expression
1 nfcv 2178 . 2  |-  F/_ x A
2 nfv 1421 . 2  |-  F/ x ps
3 elabg2.1 . 2  |-  ( x  =  A  ->  ( ps  ->  ph ) )
41, 2, 3elabgf2 9919 1  |-  ( A  e.  V  ->  ( ps  ->  A  e.  {
x  |  ph }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    e. wcel 1393   {cab 2026
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559
This theorem is referenced by: (None)
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