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Theorem elabgf1 13016
Description: One implication of elabgf 2826. (Contributed by BJ, 21-Nov-2019.)
Hypotheses
Ref Expression
elabgf1.nf1  |-  F/_ x A
elabgf1.nf2  |-  F/ x ps
elabgf1.1  |-  ( x  =  A  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
elabgf1  |-  ( A  e.  { x  | 
ph }  ->  ps )

Proof of Theorem elabgf1
StepHypRef Expression
1 elabgf1.nf1 . . 3  |-  F/_ x A
2 elabgf1.nf2 . . 3  |-  F/ x ps
31, 2elabgft1 13015 . 2  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  {
x  |  ph }  ->  ps ) )
4 elabgf1.1 . 2  |-  ( x  =  A  ->  ( ph  ->  ps ) )
53, 4mpg 1427 1  |-  ( A  e.  { x  | 
ph }  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   F/wnf 1436    e. wcel 1480   {cab 2125   F/_wnfc 2268
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:  elabf1  13018
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