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Theorem vtocl2gaf 2674
Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.)
Hypotheses
Ref Expression
vtocl2gaf.a  |-  F/_ x A
vtocl2gaf.b  |-  F/_ y A
vtocl2gaf.c  |-  F/_ y B
vtocl2gaf.1  |-  F/ x ps
vtocl2gaf.2  |-  F/ y ch
vtocl2gaf.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
vtocl2gaf.4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
vtocl2gaf.5  |-  ( ( x  e.  C  /\  y  e.  D )  ->  ph )
Assertion
Ref Expression
vtocl2gaf  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ch )
Distinct variable groups:    x, y, C   
x, D, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    ch( x, y)    A( x, y)    B( x, y)

Proof of Theorem vtocl2gaf
StepHypRef Expression
1 vtocl2gaf.a . . 3  |-  F/_ x A
2 vtocl2gaf.b . . 3  |-  F/_ y A
3 vtocl2gaf.c . . 3  |-  F/_ y B
41nfel1 2233 . . . . 5  |-  F/ x  A  e.  C
5 nfv 1462 . . . . 5  |-  F/ x  y  e.  D
64, 5nfan 1498 . . . 4  |-  F/ x
( A  e.  C  /\  y  e.  D
)
7 vtocl2gaf.1 . . . 4  |-  F/ x ps
86, 7nfim 1505 . . 3  |-  F/ x
( ( A  e.  C  /\  y  e.  D )  ->  ps )
92nfel1 2233 . . . . 5  |-  F/ y  A  e.  C
103nfel1 2233 . . . . 5  |-  F/ y  B  e.  D
119, 10nfan 1498 . . . 4  |-  F/ y ( A  e.  C  /\  B  e.  D
)
12 vtocl2gaf.2 . . . 4  |-  F/ y ch
1311, 12nfim 1505 . . 3  |-  F/ y ( ( A  e.  C  /\  B  e.  D )  ->  ch )
14 eleq1 2145 . . . . 5  |-  ( x  =  A  ->  (
x  e.  C  <->  A  e.  C ) )
1514anbi1d 453 . . . 4  |-  ( x  =  A  ->  (
( x  e.  C  /\  y  e.  D
)  <->  ( A  e.  C  /\  y  e.  D ) ) )
16 vtocl2gaf.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
1715, 16imbi12d 232 . . 3  |-  ( x  =  A  ->  (
( ( x  e.  C  /\  y  e.  D )  ->  ph )  <->  ( ( A  e.  C  /\  y  e.  D
)  ->  ps )
) )
18 eleq1 2145 . . . . 5  |-  ( y  =  B  ->  (
y  e.  D  <->  B  e.  D ) )
1918anbi2d 452 . . . 4  |-  ( y  =  B  ->  (
( A  e.  C  /\  y  e.  D
)  <->  ( A  e.  C  /\  B  e.  D ) ) )
20 vtocl2gaf.4 . . . 4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
2119, 20imbi12d 232 . . 3  |-  ( y  =  B  ->  (
( ( A  e.  C  /\  y  e.  D )  ->  ps ) 
<->  ( ( A  e.  C  /\  B  e.  D )  ->  ch ) ) )
22 vtocl2gaf.5 . . 3  |-  ( ( x  e.  C  /\  y  e.  D )  ->  ph )
231, 2, 3, 8, 13, 17, 21, 22vtocl2gf 2669 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( A  e.  C  /\  B  e.  D )  ->  ch ) )
2423pm2.43i 48 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   F/wnf 1390    e. wcel 1434   F/_wnfc 2210
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:  vtocl2ga  2675  ovmpt2s  5675  ov2gf  5676  ovi3  5688
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