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Theorem vtocl3gaf 2854
Description: Implicit substitution of 3 classes for 3 set variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)
Hypotheses
Ref Expression
vtocl3gaf.a  |-  F/_ x A
vtocl3gaf.b  |-  F/_ y A
vtocl3gaf.c  |-  F/_ z A
vtocl3gaf.d  |-  F/_ y B
vtocl3gaf.e  |-  F/_ z B
vtocl3gaf.f  |-  F/_ z C
vtocl3gaf.1  |-  F/ x ps
vtocl3gaf.2  |-  F/ y ch
vtocl3gaf.3  |-  F/ z th
vtocl3gaf.4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
vtocl3gaf.5  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
vtocl3gaf.6  |-  ( z  =  C  ->  ( ch 
<->  th ) )
vtocl3gaf.7  |-  ( ( x  e.  R  /\  y  e.  S  /\  z  e.  T )  ->  ph )
Assertion
Ref Expression
vtocl3gaf  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T )  ->  th )
Distinct variable groups:    x, y,
z, R    x, S, y, z    x, T, y, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)    ch( x, y, z)    th( x, y, z)    A( x, y, z)    B( x, y, z)    C( x, y, z)

Proof of Theorem vtocl3gaf
StepHypRef Expression
1 vtocl3gaf.a . . 3  |-  F/_ x A
2 vtocl3gaf.b . . 3  |-  F/_ y A
3 vtocl3gaf.c . . 3  |-  F/_ z A
4 vtocl3gaf.d . . 3  |-  F/_ y B
5 vtocl3gaf.e . . 3  |-  F/_ z B
6 vtocl3gaf.f . . 3  |-  F/_ z C
71nfel1 2431 . . . . 5  |-  F/ x  A  e.  R
8 nfv 1607 . . . . 5  |-  F/ x  y  e.  S
9 nfv 1607 . . . . 5  |-  F/ x  z  e.  T
107, 8, 9nf3an 1776 . . . 4  |-  F/ x
( A  e.  R  /\  y  e.  S  /\  z  e.  T
)
11 vtocl3gaf.1 . . . 4  |-  F/ x ps
1210, 11nfim 1771 . . 3  |-  F/ x
( ( A  e.  R  /\  y  e.  S  /\  z  e.  T )  ->  ps )
132nfel1 2431 . . . . 5  |-  F/ y  A  e.  R
144nfel1 2431 . . . . 5  |-  F/ y  B  e.  S
15 nfv 1607 . . . . 5  |-  F/ y  z  e.  T
1613, 14, 15nf3an 1776 . . . 4  |-  F/ y ( A  e.  R  /\  B  e.  S  /\  z  e.  T
)
17 vtocl3gaf.2 . . . 4  |-  F/ y ch
1816, 17nfim 1771 . . 3  |-  F/ y ( ( A  e.  R  /\  B  e.  S  /\  z  e.  T )  ->  ch )
193nfel1 2431 . . . . 5  |-  F/ z  A  e.  R
205nfel1 2431 . . . . 5  |-  F/ z  B  e.  S
216nfel1 2431 . . . . 5  |-  F/ z  C  e.  T
2219, 20, 21nf3an 1776 . . . 4  |-  F/ z ( A  e.  R  /\  B  e.  S  /\  C  e.  T
)
23 vtocl3gaf.3 . . . 4  |-  F/ z th
2422, 23nfim 1771 . . 3  |-  F/ z ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T )  ->  th )
25 eleq1 2345 . . . . 5  |-  ( x  =  A  ->  (
x  e.  R  <->  A  e.  R ) )
26253anbi1d 1256 . . . 4  |-  ( x  =  A  ->  (
( x  e.  R  /\  y  e.  S  /\  z  e.  T
)  <->  ( A  e.  R  /\  y  e.  S  /\  z  e.  T ) ) )
27 vtocl3gaf.4 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
2826, 27imbi12d 311 . . 3  |-  ( x  =  A  ->  (
( ( x  e.  R  /\  y  e.  S  /\  z  e.  T )  ->  ph )  <->  ( ( A  e.  R  /\  y  e.  S  /\  z  e.  T
)  ->  ps )
) )
29 eleq1 2345 . . . . 5  |-  ( y  =  B  ->  (
y  e.  S  <->  B  e.  S ) )
30293anbi2d 1257 . . . 4  |-  ( y  =  B  ->  (
( A  e.  R  /\  y  e.  S  /\  z  e.  T
)  <->  ( A  e.  R  /\  B  e.  S  /\  z  e.  T ) ) )
31 vtocl3gaf.5 . . . 4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
3230, 31imbi12d 311 . . 3  |-  ( y  =  B  ->  (
( ( A  e.  R  /\  y  e.  S  /\  z  e.  T )  ->  ps ) 
<->  ( ( A  e.  R  /\  B  e.  S  /\  z  e.  T )  ->  ch ) ) )
33 eleq1 2345 . . . . 5  |-  ( z  =  C  ->  (
z  e.  T  <->  C  e.  T ) )
34333anbi3d 1258 . . . 4  |-  ( z  =  C  ->  (
( A  e.  R  /\  B  e.  S  /\  z  e.  T
)  <->  ( A  e.  R  /\  B  e.  S  /\  C  e.  T ) ) )
35 vtocl3gaf.6 . . . 4  |-  ( z  =  C  ->  ( ch 
<->  th ) )
3634, 35imbi12d 311 . . 3  |-  ( z  =  C  ->  (
( ( A  e.  R  /\  B  e.  S  /\  z  e.  T )  ->  ch ) 
<->  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T )  ->  th )
) )
37 vtocl3gaf.7 . . 3  |-  ( ( x  e.  R  /\  y  e.  S  /\  z  e.  T )  ->  ph )
381, 2, 3, 4, 5, 6, 12, 18, 24, 28, 32, 36, 37vtocl3gf 2848 . 2  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T )  ->  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T )  ->  th )
)
3938pm2.43i 43 1  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  T )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934   F/wnf 1533    = wceq 1625    e. wcel 1686   F/_wnfc 2408
This theorem is referenced by:  vtocl3ga  2855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792
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