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Theorem elabf2 14761
Description: One implication of elabf 2892. (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 2329 . . 3  |-  F/_ x A
3 elabf2.nf . . 3  |-  F/ x ps
4 elabf2.1 . . 3  |-  ( x  =  A  ->  ( ps  ->  ph ) )
52, 3, 4elabgf2 14759 . 2  |-  ( A  e.  _V  ->  ( ps  ->  A  e.  {
x  |  ph }
) )
61, 5ax-mp 5 1  |-  ( ps 
->  A  e.  { x  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363   F/wnf 1470    e. wcel 2158   {cab 2173   _Vcvv 2749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751
This theorem is referenced by:  elab2a  14763  bj-bdfindis  14926
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