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Theorem vtoclegft 2730
Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2731.) (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 2672 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
2 exim 1561 . . . 4  |-  ( A. x ( x  =  A  ->  ph )  -> 
( E. x  x  =  A  ->  E. x ph ) )
31, 2mpan9 277 . . 3  |-  ( ( A  e.  B  /\  A. x ( x  =  A  ->  ph ) )  ->  E. x ph )
433adant2 983 . 2  |-  ( ( A  e.  B  /\  F/ x ph  /\  A. x ( x  =  A  ->  ph ) )  ->  E. x ph )
5 19.9t 1604 . . 3  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
653ad2ant2 986 . 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 945   A.wal 1312    = wceq 1314   F/wnf 1419   E.wex 1451    e. wcel 1463
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 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2660
This theorem is referenced by: (None)
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