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Theorem bj-vtoclgft 9914
Description: Weakening two hypotheses of vtoclgf 2612. (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 2566 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 bj-vtoclgf.nf1 . . . 4  |-  F/_ x A
32issetf 2562 . . 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 9909 . . 3  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( E. x  x  =  A  ->  ps ) )
73, 6syl5bi 141 . 2  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  _V  ->  ps ) )
81, 7syl5 28 1  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( A  e.  V  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1241    = wceq 1243   F/wnf 1349   E.wex 1381    e. wcel 1393   F/_wnfc 2165   _Vcvv 2557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559
This theorem is referenced by:  bj-vtoclgf  9915  elabgft1  9917  bj-rspgt  9925
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