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Theorem brcogw 4605
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 simpl1 946 . 2 (((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → 𝐴𝑉)
2 simpl2 947 . 2 (((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → 𝐵𝑊)
3 breq2 3849 . . . . . 6 (𝑥 = 𝑋 → (𝐴𝐷𝑥𝐴𝐷𝑋))
4 breq1 3848 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐶𝐵𝑋𝐶𝐵))
53, 4anbi12d 457 . . . . 5 (𝑥 = 𝑋 → ((𝐴𝐷𝑥𝑥𝐶𝐵) ↔ (𝐴𝐷𝑋𝑋𝐶𝐵)))
65spcegv 2707 . . . 4 (𝑋𝑍 → ((𝐴𝐷𝑋𝑋𝐶𝐵) → ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
76imp 122 . . 3 ((𝑋𝑍 ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
873ad2antl3 1107 . 2 (((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
9 brcog 4603 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
109biimpar 291 . 2 (((𝐴𝑉𝐵𝑊) ∧ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)) → 𝐴(𝐶𝐷)𝐵)
111, 2, 8, 10syl21anc 1173 1 (((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → 𝐴(𝐶𝐷)𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 924   = wceq 1289  wex 1426  wcel 1438   class class class wbr 3845  ccom 4442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-co 4447
This theorem is referenced by: (None)
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