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Theorem elabgf1 11022
Description: One implication of elabgf 2746. (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 11021 . 2  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  {
x  |  ph }  ->  ps ) )
4 elabgf1.1 . 2  |-  ( x  =  A  ->  ( ph  ->  ps ) )
53, 4mpg 1381 1  |-  ( A  e.  { x  | 
ph }  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   F/wnf 1390    e. wcel 1434   {cab 2069   F/_wnfc 2210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614
This theorem is referenced by:  elabf1  11024
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