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Theorem bj-vtoclgft 14149
Description: Weakening two hypotheses of vtoclgf 2795. (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 2748 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 bj-vtoclgf.nf1 . . . 4  |-  F/_ x A
32issetf 2744 . . 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 14143 . . 3  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( E. x  x  =  A  ->  ps ) )
73, 6biimtrid 152 . 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 1351    = wceq 1353   F/wnf 1460   E.wex 1492    e. wcel 2148   F/_wnfc 2306   _Vcvv 2737
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739
This theorem is referenced by:  bj-vtoclgf  14150  elabgft1  14152  bj-rspgt  14160
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