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Theorem vtoclegft 2784
Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2785.) (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 2726 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
2 exim 1579 . . . 4  |-  ( A. x ( x  =  A  ->  ph )  -> 
( E. x  x  =  A  ->  E. x ph ) )
31, 2mpan9 279 . . 3  |-  ( ( A  e.  B  /\  A. x ( x  =  A  ->  ph ) )  ->  E. x ph )
433adant2 1001 . 2  |-  ( ( A  e.  B  /\  F/ x ph  /\  A. x ( x  =  A  ->  ph ) )  ->  E. x ph )
5 19.9t 1622 . . 3  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
653ad2ant2 1004 . 2  |-  ( ( A  e.  B  /\  F/ x ph  /\  A. x ( x  =  A  ->  ph ) )  ->  ( E. x ph 
<-> 
ph ) )
74, 6mpbid 146 1  |-  ( ( A  e.  B  /\  F/ x ph  /\  A. x ( x  =  A  ->  ph ) )  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    /\ w3a 963   A.wal 1333    = wceq 1335   F/wnf 1440   E.wex 1472    e. wcel 2128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714
This theorem is referenced by: (None)
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