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Theorem vtoclgaf 3114
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 2608 . . . 4  |-  F/ x  A  e.  B
3 vtoclgaf.2 . . . 4  |-  F/ x ps
42, 3nfim 2005 . . 3  |-  F/ x
( A  e.  B  ->  ps )
5 eleq1 2519 . . . 4  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
6 vtoclgaf.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
75, 6imbi12d 322 . . 3  |-  ( x  =  A  ->  (
( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps )
) )
8 vtoclgaf.4 . . 3  |-  ( x  e.  B  ->  ph )
91, 4, 7, 8vtoclgf 3107 . 2  |-  ( A  e.  B  ->  ( A  e.  B  ->  ps ) )
109pm2.43i 49 1  |-  ( A  e.  B  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    = wceq 1446   F/wnf 1669    e. wcel 1889   F/_wnfc 2581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-v 3049
This theorem is referenced by:  vtoclga  3115  ssiun2s  4325  fvmptss  5963  fvmptf  5971  fmptco  6061  tfis  6686  inar1  9205  sumss  13802  fprodn0  14045  prmind2  14647  lss1d  18198  itg2splitlem  22718  dgrle  23209  cnlnadjlem5  27736  poimirlem25  31977  stoweidlem26  37896  iunopeqop  39015
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