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Theorem elabgf1 9918
Description: One implication of elabgf 2685. (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 9917 . 2  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  {
x  |  ph }  ->  ps ) )
4 elabgf1.1 . 2  |-  ( x  =  A  ->  ( ph  ->  ps ) )
53, 4mpg 1340 1  |-  ( A  e.  { x  | 
ph }  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243   F/wnf 1349    e. wcel 1393   {cab 2026   F/_wnfc 2165
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:  elabf1  9920
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