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Theorem brcoels 36300
Description: 𝐵 and 𝐶 are coelements : a binary relation. (Contributed by Peter Mazsa, 14-Jan-2020.) (Revised by Peter Mazsa, 5-Oct-2021.)
Assertion
Ref Expression
brcoels ((𝐵𝑉𝐶𝑊) → (𝐵𝐴𝐶 ↔ ∃𝑢𝐴 (𝐵𝑢𝐶𝑢)))
Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝐶
Allowed substitution hints:   𝑉(𝑢)   𝑊(𝑢)

Proof of Theorem brcoels
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2825 . . . 4 (𝑥 = 𝐵 → (𝑥𝑢𝐵𝑢))
2 eleq1 2825 . . . 4 (𝑦 = 𝐶 → (𝑦𝑢𝐶𝑢))
31, 2bi2anan9 639 . . 3 ((𝑥 = 𝐵𝑦 = 𝐶) → ((𝑥𝑢𝑦𝑢) ↔ (𝐵𝑢𝐶𝑢)))
43rexbidv 3221 . 2 ((𝑥 = 𝐵𝑦 = 𝐶) → (∃𝑢𝐴 (𝑥𝑢𝑦𝑢) ↔ ∃𝑢𝐴 (𝐵𝑢𝐶𝑢)))
5 dfcoels 36295 . 2 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
64, 5brabga 5420 1 ((𝐵𝑉𝐶𝑊) → (𝐵𝐴𝐶 ↔ ∃𝑢𝐴 (𝐵𝑢𝐶𝑢)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wrex 3062   class class class wbr 5058  ccoels 36076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5197  ax-nul 5204  ax-pr 5327
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3415  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-sn 4547  df-pr 4549  df-op 4553  df-br 5059  df-opab 5121  df-eprel 5465  df-xp 5562  df-rel 5563  df-cnv 5564  df-res 5568  df-coss 36279  df-coels 36280
This theorem is referenced by:  erim2  36531
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