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Theorem brcogw 4447
Description: Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
Assertion
Ref Expression
brcogw (((A 𝑉 B 𝑊 𝑋 𝑍) (A𝐷𝑋 𝑋𝐶B)) → A(𝐶𝐷)B)

Proof of Theorem brcogw
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 simpl1 906 . 2 (((A 𝑉 B 𝑊 𝑋 𝑍) (A𝐷𝑋 𝑋𝐶B)) → A 𝑉)
2 simpl2 907 . 2 (((A 𝑉 B 𝑊 𝑋 𝑍) (A𝐷𝑋 𝑋𝐶B)) → B 𝑊)
3 breq2 3759 . . . . . 6 (x = 𝑋 → (A𝐷xA𝐷𝑋))
4 breq1 3758 . . . . . 6 (x = 𝑋 → (x𝐶B𝑋𝐶B))
53, 4anbi12d 442 . . . . 5 (x = 𝑋 → ((A𝐷x x𝐶B) ↔ (A𝐷𝑋 𝑋𝐶B)))
65spcegv 2635 . . . 4 (𝑋 𝑍 → ((A𝐷𝑋 𝑋𝐶B) → x(A𝐷x x𝐶B)))
76imp 115 . . 3 ((𝑋 𝑍 (A𝐷𝑋 𝑋𝐶B)) → x(A𝐷x x𝐶B))
873ad2antl3 1067 . 2 (((A 𝑉 B 𝑊 𝑋 𝑍) (A𝐷𝑋 𝑋𝐶B)) → x(A𝐷x x𝐶B))
9 brcog 4445 . . 3 ((A 𝑉 B 𝑊) → (A(𝐶𝐷)Bx(A𝐷x x𝐶B)))
109biimpar 281 . 2 (((A 𝑉 B 𝑊) x(A𝐷x x𝐶B)) → A(𝐶𝐷)B)
111, 2, 8, 10syl21anc 1133 1 (((A 𝑉 B 𝑊 𝑋 𝑍) (A𝐷𝑋 𝑋𝐶B)) → A(𝐶𝐷)B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242  wex 1378   wcel 1390   class class class wbr 3755  ccom 4292
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-co 4297
This theorem is referenced by: (None)
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