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Theorem vtocl2ga 2675
Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)
Hypotheses
Ref Expression
vtocl2ga.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
vtocl2ga.2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
vtocl2ga.3  |-  ( ( x  e.  C  /\  y  e.  D )  ->  ph )
Assertion
Ref Expression
vtocl2ga  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ch )
Distinct variable groups:    x, y, A   
y, B    x, C, y    x, D, y    ps, x    ch, y
Allowed substitution hints:    ph( x, y)    ps( y)    ch( x)    B( x)

Proof of Theorem vtocl2ga
StepHypRef Expression
1 nfcv 2223 . 2  |-  F/_ x A
2 nfcv 2223 . 2  |-  F/_ y A
3 nfcv 2223 . 2  |-  F/_ y B
4 nfv 1462 . 2  |-  F/ x ps
5 nfv 1462 . 2  |-  F/ y ch
6 vtocl2ga.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
7 vtocl2ga.2 . 2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
8 vtocl2ga.3 . 2  |-  ( ( x  e.  C  /\  y  e.  D )  ->  ph )
91, 2, 3, 4, 5, 6, 7, 8vtocl2gaf 2674 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> 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:  caovcan  5716  genipv  6813
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