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Theorem vtoclegft 2691
Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2692.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
Assertion
Ref Expression
vtoclegft  |-  ( ( A  e.  B  /\  F/ x ph  /\  A. x ( x  =  A  ->  ph ) )  ->  ph )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem vtoclegft
StepHypRef Expression
1 elisset 2633 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
2 exim 1535 . . . 4  |-  ( A. x ( x  =  A  ->  ph )  -> 
( E. x  x  =  A  ->  E. x ph ) )
31, 2mpan9 275 . . 3  |-  ( ( A  e.  B  /\  A. x ( x  =  A  ->  ph ) )  ->  E. x ph )
433adant2 962 . 2  |-  ( ( A  e.  B  /\  F/ x ph  /\  A. x ( x  =  A  ->  ph ) )  ->  E. x ph )
5 19.9t 1578 . . 3  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
653ad2ant2 965 . 2  |-  ( ( A  e.  B  /\  F/ x ph  /\  A. x ( x  =  A  ->  ph ) )  ->  ( E. x ph 
<-> 
ph ) )
74, 6mpbid 145 1  |-  ( ( A  e.  B  /\  F/ x ph  /\  A. x ( x  =  A  ->  ph ) )  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    /\ w3a 924   A.wal 1287    = wceq 1289   F/wnf 1394   E.wex 1426    e. wcel 1438
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-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by: (None)
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