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Theorem vtocl2g 2676
Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.)
Hypotheses
Ref Expression
vtocl2g.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
vtocl2g.2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
vtocl2g.3  |-  ph
Assertion
Ref Expression
vtocl2g  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ch )
Distinct variable groups:    x, A    y, A    y, B    ps, x    ch, y
Allowed substitution hints:    ph( x, y)    ps( y)    ch( x)    B( x)    V( x, y)    W( x, y)

Proof of Theorem vtocl2g
StepHypRef Expression
1 nfcv 2225 . 2  |-  F/_ x A
2 nfcv 2225 . 2  |-  F/_ y A
3 nfcv 2225 . 2  |-  F/_ y B
4 nfv 1464 . 2  |-  F/ x ps
5 nfv 1464 . 2  |-  F/ y ch
6 vtocl2g.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
7 vtocl2g.2 . 2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
8 vtocl2g.3 . 2  |-  ph
91, 2, 3, 4, 5, 6, 7, 8vtocl2gf 2674 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> 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:  uniprg  3651  intprg  3704  opthg  4039  opelopabsb  4061  unexb  4241  vtoclr  4454  elimasng  4767  cnvsng  4882  funopg  5013  f1osng  5257  fsng  5433  fvsng  5456  op1stg  5878  op2ndg  5879  xpsneng  6490  xpcomeng  6496  bdunexb  11249  bj-unexg  11250
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