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Theorem vtoclbg 2668
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994.)
Hypotheses
Ref Expression
vtoclbg.1  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
vtoclbg.2  |-  ( x  =  A  ->  ( ps 
<->  th ) )
vtoclbg.3  |-  ( ph  <->  ps )
Assertion
Ref Expression
vtoclbg  |-  ( A  e.  V  ->  ( ch 
<->  th ) )
Distinct variable groups:    x, A    ch, x    th, x
Allowed substitution hints:    ph( x)    ps( x)    V( x)

Proof of Theorem vtoclbg
StepHypRef Expression
1 vtoclbg.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
2 vtoclbg.2 . . 3  |-  ( x  =  A  ->  ( ps 
<->  th ) )
31, 2bibi12d 233 . 2  |-  ( x  =  A  ->  (
( ph  <->  ps )  <->  ( ch  <->  th ) ) )
4 vtoclbg.3 . 2  |-  ( ph  <->  ps )
53, 4vtoclg 2667 1  |-  ( A  e.  V  ->  ( ch 
<->  th ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612
This theorem is referenced by:  pm13.183  2740  sbc8g  2831  sbcco  2845  sbc5  2847  sbcie2g  2856  eqsbc3  2862  sbcng  2863  sbcimg  2864  sbcan  2865  sbcang  2866  sbcor  2867  sbcorg  2868  sbcbig  2869  sbcal  2874  sbcalg  2875  sbcex2  2876  sbcexg  2877  sbcel1v  2885  sbcralg  2901  sbcreug  2903  sbcel12g  2930  sbceqg  2931  csbiebg  2954  elpwg  3408  snssg  3541  preq12bg  3585  elintg  3664  elintrabg  3669  sbcbrg  3854  opelresg  4667  domeng  6320
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