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Theorem bj-vtoclgft 15005
Description: Weakening two hypotheses of vtoclgf 2810. (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 2763 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 bj-vtoclgf.nf1 . . . 4  |-  F/_ x A
32issetf 2759 . . 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 14999 . . 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 1362    = wceq 1364   F/wnf 1471   E.wex 1503    e. wcel 2160   F/_wnfc 2319   _Vcvv 2752
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754
This theorem is referenced by:  bj-vtoclgf  15006  elabgft1  15008  bj-rspgt  15016
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