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Theorem vtoclg1f 2796
Description: Version of vtoclgf 2795 with one nonfreeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-11 1506 and ax-13 2150. (Contributed by BJ, 1-May-2019.)
Hypotheses
Ref Expression
vtoclg1f.nf  |-  F/ x ps
vtoclg1f.maj  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
vtoclg1f.min  |-  ph
Assertion
Ref Expression
vtoclg1f  |-  ( A  e.  V  ->  ps )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)    V( x)

Proof of Theorem vtoclg1f
StepHypRef Expression
1 elex 2748 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 isset 2743 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
3 vtoclg1f.nf . . . 4  |-  F/ x ps
4 vtoclg1f.min . . . . 5  |-  ph
5 vtoclg1f.maj . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
64, 5mpbii 148 . . . 4  |-  ( x  =  A  ->  ps )
73, 6exlimi 1594 . . 3  |-  ( E. x  x  =  A  ->  ps )
82, 7sylbi 121 . 2  |-  ( A  e.  _V  ->  ps )
91, 8syl 14 1  |-  ( A  e.  V  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353   F/wnf 1460   E.wex 1492    e. wcel 2148   _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-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2739
This theorem is referenced by:  opeliunxp2f  6238  summodclem2a  11384  fprodsplit1f  11637
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