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Theorem brcoels 35560
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 2897 . . . 4 (𝑥 = 𝐵 → (𝑥𝑢𝐵𝑢))
2 eleq1 2897 . . . 4 (𝑦 = 𝐶 → (𝑦𝑢𝐶𝑢))
31, 2bi2anan9 635 . . 3 ((𝑥 = 𝐵𝑦 = 𝐶) → ((𝑥𝑢𝑦𝑢) ↔ (𝐵𝑢𝐶𝑢)))
43rexbidv 3294 . 2 ((𝑥 = 𝐵𝑦 = 𝐶) → (∃𝑢𝐴 (𝑥𝑢𝑦𝑢) ↔ ∃𝑢𝐴 (𝐵𝑢𝐶𝑢)))
5 dfcoels 35555 . 2 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
64, 5brabga 5412 1 ((𝐵𝑉𝐶𝑊) → (𝐵𝐴𝐶 ↔ ∃𝑢𝐴 (𝐵𝑢𝐶𝑢)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wrex 3136   class class class wbr 5057  ccoels 35335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-eprel 5458  df-xp 5554  df-rel 5555  df-cnv 5556  df-res 5560  df-coss 35539  df-coels 35540
This theorem is referenced by:  erim2  35791
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