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Theorem brcoels 38034
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 2813 . . . 4 (𝑥 = 𝐵 → (𝑥𝑢𝐵𝑢))
2 eleq1 2813 . . . 4 (𝑦 = 𝐶 → (𝑦𝑢𝐶𝑢))
31, 2bi2anan9 636 . . 3 ((𝑥 = 𝐵𝑦 = 𝐶) → ((𝑥𝑢𝑦𝑢) ↔ (𝐵𝑢𝐶𝑢)))
43rexbidv 3168 . 2 ((𝑥 = 𝐵𝑦 = 𝐶) → (∃𝑢𝐴 (𝑥𝑢𝑦𝑢) ↔ ∃𝑢𝐴 (𝐵𝑢𝐶𝑢)))
5 dfcoels 38029 . 2 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
64, 5brabga 5536 1 ((𝐵𝑉𝐶𝑊) → (𝐵𝐴𝐶 ↔ ∃𝑢𝐴 (𝐵𝑢𝐶𝑢)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wrex 3059   class class class wbr 5149  ccoels 37777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-eprel 5582  df-xp 5684  df-rel 5685  df-cnv 5686  df-res 5690  df-coss 38010  df-coels 38011
This theorem is referenced by:  erimeq2  38277
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