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Theorem elabf2 13019
Description: One implication of elabf 2827. (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 2281 . . 3  |-  F/_ x A
3 elabf2.nf . . 3  |-  F/ x ps
4 elabf2.1 . . 3  |-  ( x  =  A  ->  ( ps  ->  ph ) )
52, 3, 4elabgf2 13017 . 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 1331   F/wnf 1436    e. wcel 1480   {cab 2125   _Vcvv 2686
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:  elab2a  13021  bj-bdfindis  13175
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