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Theorem vtoclga 2747
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.)
Hypotheses
Ref Expression
vtoclga.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
vtoclga.2  |-  ( x  e.  B  ->  ph )
Assertion
Ref Expression
vtoclga  |-  ( A  e.  B  ->  ps )
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem vtoclga
StepHypRef Expression
1 nfcv 2279 . 2  |-  F/_ x A
2 nfv 1508 . 2  |-  F/ x ps
3 vtoclga.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
4 vtoclga.2 . 2  |-  ( x  e.  B  ->  ph )
51, 2, 3, 4vtoclgaf 2746 1  |-  ( A  e.  B  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    e. wcel 1480
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 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683
This theorem is referenced by:  vtoclri  2756  ssuni  3753  ordtriexmid  4432  onsucsssucexmid  4437  tfis3  4495  fvmpt3  5493  fvmptssdm  5498  fnressn  5599  fressnfv  5600  caovord  5935  caovimo  5957  tfrlem1  6198  nnacl  6369  nnmcl  6370  nnacom  6373  nnaass  6374  nndi  6375  nnmass  6376  nnmsucr  6377  nnmcom  6378  nnsucsssuc  6381  nntri3or  6382  nnaordi  6397  nnaword  6400  nnmordi  6405  nnaordex  6416  ixpfn  6591  findcard  6775  findcard2  6776  findcard2s  6777  exmidomni  7007  indpi  7143  prarloclem3  7298  uzind4s2  9379  cnref1o  9433  frec2uzrdg  10175  expcl2lemap  10298  seq3coll  10578  climub  11106  climserle  11107  fsum3cvg  11139  summodclem2a  11143  prodfap0  11307  prodfrecap  11308  fproddccvg  11334  alginv  11717  algcvg  11718  algcvga  11721  algfx  11722  prmind2  11790
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