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Theorem brcogw 5004
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 986 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  A  e.  V )
2 simpl2 987 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  B  e.  W )
3 breq2 4293 . . . . . 6  |-  ( x  =  X  ->  ( A D x  <->  A D X ) )
4 breq1 4292 . . . . . 6  |-  ( x  =  X  ->  (
x C B  <->  X C B ) )
53, 4anbi12d 705 . . . . 5  |-  ( x  =  X  ->  (
( A D x  /\  x C B )  <->  ( A D X  /\  X C B ) ) )
65spcegv 3055 . . . 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 1147 . 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 5002 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
109biimpar 482 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  E. x
( A D x  /\  x C B ) )  ->  A
( C  o.  D
) B )
111, 2, 8, 10syl21anc 1212 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 960    = wceq 1364   E.wex 1591    e. wcel 1761   class class class wbr 4289    o. ccom 4840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-co 4845
This theorem is referenced by:  utop2nei  19784  utop3cls  19785
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