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Theorem elabf2 11025
Description: One implication of elabf 2747. (Contributed by BJ, 21-Nov-2019.)
Hypotheses
Ref Expression
elabf2.nf  |-  F/ x ps
elabf2.s  |-  A  e. 
_V
elabf2.1  |-  ( x  =  A  ->  ( ps  ->  ph ) )
Assertion
Ref Expression
elabf2  |-  ( ps 
->  A  e.  { x  |  ph } )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem elabf2
StepHypRef Expression
1 elabf2.s . 2  |-  A  e. 
_V
2 nfcv 2223 . . 3  |-  F/_ x A
3 elabf2.nf . . 3  |-  F/ x ps
4 elabf2.1 . . 3  |-  ( x  =  A  ->  ( ps  ->  ph ) )
52, 3, 4elabgf2 11023 . 2  |-  ( A  e.  _V  ->  ( ps  ->  A  e.  {
x  |  ph }
) )
61, 5ax-mp 7 1  |-  ( ps 
->  A  e.  { x  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   F/wnf 1390    e. wcel 1434   {cab 2069   _Vcvv 2612
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:  elab2a  11027  bj-bdfindis  11185
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