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Theorem brcoels 34812
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 2846 . . . 4 (𝑥 = 𝐵 → (𝑥𝑢𝐵𝑢))
2 eleq1 2846 . . . 4 (𝑦 = 𝐶 → (𝑦𝑢𝐶𝑢))
31, 2bi2anan9 629 . . 3 ((𝑥 = 𝐵𝑦 = 𝐶) → ((𝑥𝑢𝑦𝑢) ↔ (𝐵𝑢𝐶𝑢)))
43rexbidv 3236 . 2 ((𝑥 = 𝐵𝑦 = 𝐶) → (∃𝑢𝐴 (𝑥𝑢𝑦𝑢) ↔ ∃𝑢𝐴 (𝐵𝑢𝐶𝑢)))
5 dfcoels 34807 . 2 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
64, 5brabga 5226 1 ((𝐵𝑉𝐶𝑊) → (𝐵𝐴𝐶 ↔ ∃𝑢𝐴 (𝐵𝑢𝐶𝑢)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2106  wrex 3090   class class class wbr 4886  ccoels 34601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-eprel 5266  df-xp 5361  df-rel 5362  df-cnv 5363  df-res 5367  df-coss 34791  df-coels 34792
This theorem is referenced by: (None)
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