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Theorem vtoclri 2761
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 2484 . 2  |-  ( x  e.  B  ->  ph )
41, 3vtoclga 2752 1  |-  ( A  e.  B  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    e. wcel 1480   A.wral 2416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688
This theorem is referenced by:  ordpwsucexmid  4485  bj-nn0suc0  13253
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