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Theorem elabg2 13022
Description: One implication of elabg 2830. (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 2281 . 2  |-  F/_ x A
2 nfv 1508 . 2  |-  F/ x ps
3 elabg2.1 . 2  |-  ( x  =  A  ->  ( ps  ->  ph ) )
41, 2, 3elabgf2 13017 1  |-  ( A  e.  V  ->  ( ps  ->  A  e.  {
x  |  ph }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = 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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688
This theorem is referenced by: (None)
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