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Theorem brcogw 5825
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 1149 . 2 ((𝐴𝑉𝐵𝑊𝑋𝑍) → (𝐴𝑉𝐵𝑊))
2 breq2 5110 . . . . . 6 (𝑥 = 𝑋 → (𝐴𝐷𝑥𝐴𝐷𝑋))
3 breq1 5109 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐶𝐵𝑋𝐶𝐵))
42, 3anbi12d 632 . . . . 5 (𝑥 = 𝑋 → ((𝐴𝐷𝑥𝑥𝐶𝐵) ↔ (𝐴𝐷𝑋𝑋𝐶𝐵)))
54spcegv 3555 . . . 4 (𝑋𝑍 → ((𝐴𝐷𝑋𝑋𝐶𝐵) → ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
65imp 408 . . 3 ((𝑋𝑍 ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
763ad2antl3 1188 . 2 (((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
8 brcog 5823 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
98biimpar 479 . 2 (((𝐴𝑉𝐵𝑊) ∧ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)) → 𝐴(𝐶𝐷)𝐵)
101, 7, 9syl2an2r 684 1 (((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → 𝐴(𝐶𝐷)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107   class class class wbr 5106  ccom 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-co 5643
This theorem is referenced by:  utop2nei  23618  utop3cls  23619  iunrelexpuztr  42079  frege96d  42109  frege98d  42113
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