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Theorem brcoels 35720
 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 2899 . . . 4 (𝑥 = 𝐵 → (𝑥𝑢𝐵𝑢))
2 eleq1 2899 . . . 4 (𝑦 = 𝐶 → (𝑦𝑢𝐶𝑢))
31, 2bi2anan9 638 . . 3 ((𝑥 = 𝐵𝑦 = 𝐶) → ((𝑥𝑢𝑦𝑢) ↔ (𝐵𝑢𝐶𝑢)))
43rexbidv 3283 . 2 ((𝑥 = 𝐵𝑦 = 𝐶) → (∃𝑢𝐴 (𝑥𝑢𝑦𝑢) ↔ ∃𝑢𝐴 (𝐵𝑢𝐶𝑢)))
5 dfcoels 35715 . 2 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
64, 5brabga 5394 1 ((𝐵𝑉𝐶𝑊) → (𝐵𝐴𝐶 ↔ ∃𝑢𝐴 (𝐵𝑢𝐶𝑢)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∃wrex 3127   class class class wbr 5039   ∼ ccoels 35494 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-eprel 5438  df-xp 5534  df-rel 5535  df-cnv 5536  df-res 5540  df-coss 35699  df-coels 35700 This theorem is referenced by:  erim2  35951
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