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Theorem brcogw 4531
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 918 . 2 (((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → 𝐴𝑉)
2 simpl2 919 . 2 (((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → 𝐵𝑊)
3 breq2 3795 . . . . . 6 (𝑥 = 𝑋 → (𝐴𝐷𝑥𝐴𝐷𝑋))
4 breq1 3794 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐶𝐵𝑋𝐶𝐵))
53, 4anbi12d 450 . . . . 5 (𝑥 = 𝑋 → ((𝐴𝐷𝑥𝑥𝐶𝐵) ↔ (𝐴𝐷𝑋𝑋𝐶𝐵)))
65spcegv 2658 . . . 4 (𝑋𝑍 → ((𝐴𝐷𝑋𝑋𝐶𝐵) → ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
76imp 119 . . 3 ((𝑋𝑍 ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
873ad2antl3 1079 . 2 (((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
9 brcog 4529 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
109biimpar 285 . 2 (((𝐴𝑉𝐵𝑊) ∧ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)) → 𝐴(𝐶𝐷)𝐵)
111, 2, 8, 10syl21anc 1145 1 (((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → 𝐴(𝐶𝐷)𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 896   = wceq 1259  wex 1397  wcel 1409   class class class wbr 3791  ccom 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-co 4381
This theorem is referenced by: (None)
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