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Theorem vtoclgaf 3018
Description: Implicit substitution of a class for a set variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)
Hypotheses
Ref Expression
vtoclgaf.1  |-  F/_ x A
vtoclgaf.2  |-  F/ x ps
vtoclgaf.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
vtoclgaf.4  |-  ( x  e.  B  ->  ph )
Assertion
Ref Expression
vtoclgaf  |-  ( A  e.  B  ->  ps )
Distinct variable group:    x, B
Allowed substitution hints:    ph( x)    ps( x)    A( x)

Proof of Theorem vtoclgaf
StepHypRef Expression
1 vtoclgaf.1 . . 3  |-  F/_ x A
21nfel1 2584 . . . 4  |-  F/ x  A  e.  B
3 vtoclgaf.2 . . . 4  |-  F/ x ps
42, 3nfim 1833 . . 3  |-  F/ x
( A  e.  B  ->  ps )
5 eleq1 2498 . . . 4  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
6 vtoclgaf.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
75, 6imbi12d 313 . . 3  |-  ( x  =  A  ->  (
( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps )
) )
8 vtoclgaf.4 . . 3  |-  ( x  e.  B  ->  ph )
91, 4, 7, 8vtoclgf 3012 . 2  |-  ( A  e.  B  ->  ( A  e.  B  ->  ps ) )
109pm2.43i 46 1  |-  ( A  e.  B  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   F/wnf 1554    = wceq 1653    e. wcel 1726   F/_wnfc 2561
This theorem is referenced by:  vtoclga  3019  ssiun2s  4137  tfis  4836  fvmptss  5815  fvmptf  5823  fmptco  5903  inar1  8652  sumss  12520  prmind2  13092  lss1d  16041  itg2splitlem  19642  dgrle  20164  cnlnadjlem5  23576  fprodn0  25305  stoweidlem26  27753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960
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