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Theorem vtoclbg 2673
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 2672 1  |-  ( A  e.  V  ->  ( ch 
<->  th ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1287    e. wcel 1436
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617
This theorem is referenced by:  pm13.183  2745  sbc8g  2836  sbcco  2850  sbc5  2852  sbcie2g  2861  eqsbc3  2867  sbcng  2868  sbcimg  2869  sbcan  2870  sbcang  2871  sbcor  2872  sbcorg  2873  sbcbig  2874  sbcal  2879  sbcalg  2880  sbcex2  2881  sbcexg  2882  sbcel1v  2890  sbcralg  2906  sbcreug  2908  sbcel12g  2935  sbceqg  2936  csbiebg  2959  elpwg  3423  snssg  3558  preq12bg  3602  elintg  3681  elintrabg  3686  sbcbrg  3871  opelresg  4690  domeng  6423
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