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Theorem vtoclgaf 3172
Description: Implicit substitution of a class for a setvar 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 2635 . . . 4  |-  F/ x  A  e.  B
3 vtoclgaf.2 . . . 4  |-  F/ x ps
42, 3nfim 1921 . . 3  |-  F/ x
( A  e.  B  ->  ps )
5 eleq1 2529 . . . 4  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
6 vtoclgaf.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
75, 6imbi12d 320 . . 3  |-  ( x  =  A  ->  (
( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps )
) )
8 vtoclgaf.4 . . 3  |-  ( x  e.  B  ->  ph )
91, 4, 7, 8vtoclgf 3165 . 2  |-  ( A  e.  B  ->  ( A  e.  B  ->  ps ) )
109pm2.43i 47 1  |-  ( A  e.  B  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395   F/wnf 1617    e. wcel 1819   F/_wnfc 2605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111
This theorem is referenced by:  vtoclga  3173  ssiun2s  4376  fvmptss  5965  fvmptf  5973  fmptco  6065  tfis  6688  inar1  9170  sumss  13558  fprodn0  13795  prmind2  14240  lss1d  17736  itg2splitlem  22281  dgrle  22766  cnlnadjlem5  27117  stoweidlem26  32011
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