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Theorem vtoclegft 3165
Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3166.) (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 3104 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
2 exim 1639 . . . 4  |-  ( A. x ( x  =  A  ->  ph )  -> 
( E. x  x  =  A  ->  E. x ph ) )
31, 2mpan9 469 . . 3  |-  ( ( A  e.  B  /\  A. x ( x  =  A  ->  ph ) )  ->  E. x ph )
433adant2 1014 . 2  |-  ( ( A  e.  B  /\  F/ x ph  /\  A. x ( x  =  A  ->  ph ) )  ->  E. x ph )
5 19.9t 1874 . . 3  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
653ad2ant2 1017 . 2  |-  ( ( A  e.  B  /\  F/ x ph  /\  A. x ( x  =  A  ->  ph ) )  ->  ( E. x ph 
<-> 
ph ) )
74, 6mpbid 210 1  |-  ( ( A  e.  B  /\  F/ x ph  /\  A. x ( x  =  A  ->  ph ) )  ->  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 972   A.wal 1379    = wceq 1381   E.wex 1597   F/wnf 1601    e. wcel 1802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-12 1838  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-v 3095
This theorem is referenced by: (None)
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