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Theorem vtoclgaf 3030
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 2584 . . . 4  |-  F/ x  A  e.  B
3 vtoclgaf.2 . . . 4  |-  F/ x ps
42, 3nfim 1852 . . 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 320 . . 3  |-  ( x  =  A  ->  (
( x  e.  B  ->  ph )  <->  ( A  e.  B  ->  ps )
) )
8 vtoclgaf.4 . . 3  |-  ( x  e.  B  ->  ph )
91, 4, 7, 8vtoclgf 3023 . 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 1369   F/wnf 1589    e. wcel 1756   F/_wnfc 2561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2969
This theorem is referenced by:  vtoclga  3031  ssiun2s  4209  fvmptss  5777  fvmptf  5785  fmptco  5871  tfis  6460  inar1  8934  sumss  13193  prmind2  13766  lss1d  17021  itg2splitlem  21201  dgrle  21686  cnlnadjlem5  25426  fprodn0  27441  stoweidlem26  29774
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