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Theorem bj-vtoclgft 10870
Description: Weakening two hypotheses of vtoclgf 2667. (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 2620 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 bj-vtoclgf.nf1 . . . 4  |-  F/_ x A
32issetf 2616 . . 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 10865 . . 3  |-  ( A. x ( x  =  A  ->  ( ph  ->  ps ) )  -> 
( E. x  x  =  A  ->  ps ) )
73, 6syl5bi 150 . 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 1283    = wceq 1285   F/wnf 1390   E.wex 1422    e. wcel 1434   F/_wnfc 2210   _Vcvv 2611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2613
This theorem is referenced by:  bj-vtoclgf  10871  elabgft1  10873  bj-rspgt  10881
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