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Theorem brcogw 5106
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 991 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  A  e.  V )
2 simpl2 992 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  B  e.  W )
3 breq2 4394 . . . . . 6  |-  ( x  =  X  ->  ( A D x  <->  A D X ) )
4 breq1 4393 . . . . . 6  |-  ( x  =  X  ->  (
x C B  <->  X C B ) )
53, 4anbi12d 710 . . . . 5  |-  ( x  =  X  ->  (
( A D x  /\  x C B )  <->  ( A D X  /\  X C B ) ) )
65spcegv 3154 . . . 4  |-  ( X  e.  Z  ->  (
( A D X  /\  X C B )  ->  E. x
( A D x  /\  x C B ) ) )
76imp 429 . . 3  |-  ( ( X  e.  Z  /\  ( A D X  /\  X C B ) )  ->  E. x ( A D x  /\  x C B ) )
873ad2antl3 1152 . 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 5104 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
109biimpar 485 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  E. x
( A D x  /\  x C B ) )  ->  A
( C  o.  D
) B )
111, 2, 8, 10syl21anc 1218 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758   class class class wbr 4390    o. ccom 4942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-br 4391  df-opab 4449  df-co 4947
This theorem is referenced by:  utop2nei  19941  utop3cls  19942
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