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Theorem vtocl2gaf 3032
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 2584 . . . . 5  |-  F/ x  A  e.  C
5 nfv 1673 . . . . 5  |-  F/ x  y  e.  D
64, 5nfan 1860 . . . 4  |-  F/ x
( A  e.  C  /\  y  e.  D
)
7 vtocl2gaf.1 . . . 4  |-  F/ x ps
86, 7nfim 1852 . . 3  |-  F/ x
( ( A  e.  C  /\  y  e.  D )  ->  ps )
92nfel1 2584 . . . . 5  |-  F/ y  A  e.  C
103nfel1 2584 . . . . 5  |-  F/ y  B  e.  D
119, 10nfan 1860 . . . 4  |-  F/ y ( A  e.  C  /\  B  e.  D
)
12 vtocl2gaf.2 . . . 4  |-  F/ y ch
1311, 12nfim 1852 . . 3  |-  F/ y ( ( A  e.  C  /\  B  e.  D )  ->  ch )
14 eleq1 2498 . . . . 5  |-  ( x  =  A  ->  (
x  e.  C  <->  A  e.  C ) )
1514anbi1d 704 . . . 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 320 . . 3  |-  ( x  =  A  ->  (
( ( x  e.  C  /\  y  e.  D )  ->  ph )  <->  ( ( A  e.  C  /\  y  e.  D
)  ->  ps )
) )
18 eleq1 2498 . . . . 5  |-  ( y  =  B  ->  (
y  e.  D  <->  B  e.  D ) )
1918anbi2d 703 . . . 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 320 . . 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 3027 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( A  e.  C  /\  B  e.  D )  ->  ch ) )
2423pm2.43i 47 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   F/wnf 1589    e. wcel 1756   F/_wnfc 2561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2969
This theorem is referenced by:  vtocl2ga  3033  ovmpt2s  6209  ov2gf  6210  ov3  6222  pwfseqlem2  8818  cnmptcom  19226
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