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Theorem vtoclri 3184
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)
Hypotheses
Ref Expression
vtoclri.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
vtoclri.2  |-  A. x  e.  B  ph
Assertion
Ref Expression
vtoclri  |-  ( A  e.  B  ->  ps )
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem vtoclri
StepHypRef Expression
1 vtoclri.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
2 vtoclri.2 . . 3  |-  A. x  e.  B  ph
32rspec 2825 . 2  |-  ( x  e.  B  ->  ph )
41, 3vtoclga 3173 1  |-  ( A  e.  B  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395    e. wcel 1819   A.wral 2807
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-ral 2812  df-v 3111
This theorem is referenced by:  alephreg  8974  arch  10813  harmonicbnd  23459  harmonicbnd2  23460  ghomgrpilem1  29222  heiborlem8  30498  fourierdlem62  32133  srhmsubclem1  33004  srhmsubc  33007  srhmsubcOLDlem1  33023  srhmsubcOLD  33026
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