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Theorem brcogw 4504
Description: Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
Assertion
Ref Expression
brcogw  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  A ( C  o.  D ) B )

Proof of Theorem brcogw
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl1 907 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  A  e.  V )
2 simpl2 908 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  B  e.  W )
3 breq2 3768 . . . . . 6  |-  ( x  =  X  ->  ( A D x  <->  A D X ) )
4 breq1 3767 . . . . . 6  |-  ( x  =  X  ->  (
x C B  <->  X C B ) )
53, 4anbi12d 442 . . . . 5  |-  ( x  =  X  ->  (
( A D x  /\  x C B )  <->  ( A D X  /\  X C B ) ) )
65spcegv 2641 . . . 4  |-  ( X  e.  Z  ->  (
( A D X  /\  X C B )  ->  E. x
( A D x  /\  x C B ) ) )
76imp 115 . . 3  |-  ( ( X  e.  Z  /\  ( A D X  /\  X C B ) )  ->  E. x ( A D x  /\  x C B ) )
873ad2antl3 1068 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  E. x ( A D x  /\  x C B ) )
9 brcog 4502 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
109biimpar 281 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  E. x
( A D x  /\  x C B ) )  ->  A
( C  o.  D
) B )
111, 2, 8, 10syl21anc 1134 1  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  A ( C  o.  D ) B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    /\ w3a 885    = wceq 1243   E.wex 1381    e. wcel 1393   class class class wbr 3764    o. ccom 4349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-co 4354
This theorem is referenced by: (None)
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