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Theorem brcogw 5586
Description: Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
Assertion
Ref Expression
brcogw (((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → 𝐴(𝐶𝐷)𝐵)

Proof of Theorem brcogw
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 3simpa 1129 . 2 ((𝐴𝑉𝐵𝑊𝑋𝑍) → (𝐴𝑉𝐵𝑊))
2 breq2 4930 . . . . . 6 (𝑥 = 𝑋 → (𝐴𝐷𝑥𝐴𝐷𝑋))
3 breq1 4929 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐶𝐵𝑋𝐶𝐵))
42, 3anbi12d 622 . . . . 5 (𝑥 = 𝑋 → ((𝐴𝐷𝑥𝑥𝐶𝐵) ↔ (𝐴𝐷𝑋𝑋𝐶𝐵)))
54spcegv 3511 . . . 4 (𝑋𝑍 → ((𝐴𝐷𝑋𝑋𝐶𝐵) → ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
65imp 398 . . 3 ((𝑋𝑍 ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
763ad2antl3 1168 . 2 (((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
8 brcog 5584 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
98biimpar 470 . 2 (((𝐴𝑉𝐵𝑊) ∧ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)) → 𝐴(𝐶𝐷)𝐵)
101, 7, 9syl2an2r 673 1 (((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → 𝐴(𝐶𝐷)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1069   = wceq 1508  wex 1743  wcel 2051   class class class wbr 4926  ccom 5408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pr 5183
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-rab 3092  df-v 3412  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-br 4927  df-opab 4989  df-co 5413
This theorem is referenced by:  utop2nei  22578  utop3cls  22579  iunrelexpuztr  39461  frege96d  39491  frege98d  39495
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