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Theorem vtoclri 3193
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 2835 . 2  |-  ( x  e.  B  ->  ph )
41, 3vtoclga 3182 1  |-  ( A  e.  B  ->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   A.wral 2817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-v 3120
This theorem is referenced by:  alephreg  8969  arch  10804  harmonicbnd  23199  harmonicbnd2  23200  ghomgrpilem1  28850  heiborlem8  30241  srhmsubclem1  32386  srhmsubc  32389
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