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Theorem bj-vtoclgft 13087
Description: Weakening two hypotheses of vtoclgf 2744. (Contributed by BJ, 21-Nov-2019.)
Hypotheses
Ref Expression
bj-vtoclgf.nf1  |-  F/_ x A
bj-vtoclgf.nf2  |-  F/ x ps
bj-vtoclgf.min  |-  ( x  =  A  ->  ph )
Assertion
Ref Expression
bj-vtoclgft  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  V  ->  ps ) )

Proof of Theorem bj-vtoclgft
StepHypRef Expression
1 elex 2697 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 bj-vtoclgf.nf1 . . . 4  |-  F/_ x A
32issetf 2693 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
4 bj-vtoclgf.nf2 . . . 4  |-  F/ x ps
5 bj-vtoclgf.min . . . 4  |-  ( x  =  A  ->  ph )
64, 5bj-exlimmp 13081 . . 3  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( E. x  x  =  A  ->  ps ) )
73, 6syl5bi 151 . 2  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  _V  ->  ps ) )
81, 7syl5 32 1  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  V  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1329    = wceq 1331   F/wnf 1436   E.wex 1468    e. wcel 1480   F/_wnfc 2268   _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:  bj-vtoclgf  13088  elabgft1  13090  bj-rspgt  13098
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