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Theorem brcodir 6123
Description: Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)
Assertion
Ref Expression
brcodir ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧𝐵𝑅𝑧)))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑧,𝑅   𝑧,𝑉   𝑧,𝑊

Proof of Theorem brcodir
StepHypRef Expression
1 brcog 5865 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧𝑧𝑅𝐵)))
2 vex 3466 . . . . . 6 𝑧 ∈ V
3 brcnvg 5878 . . . . . 6 ((𝑧 ∈ V ∧ 𝐵𝑊) → (𝑧𝑅𝐵𝐵𝑅𝑧))
42, 3mpan 688 . . . . 5 (𝐵𝑊 → (𝑧𝑅𝐵𝐵𝑅𝑧))
54anbi2d 628 . . . 4 (𝐵𝑊 → ((𝐴𝑅𝑧𝑧𝑅𝐵) ↔ (𝐴𝑅𝑧𝐵𝑅𝑧)))
65adantl 480 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑅𝑧𝑧𝑅𝐵) ↔ (𝐴𝑅𝑧𝐵𝑅𝑧)))
76exbidv 1917 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑧(𝐴𝑅𝑧𝑧𝑅𝐵) ↔ ∃𝑧(𝐴𝑅𝑧𝐵𝑅𝑧)))
81, 7bitrd 278 1 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧𝐵𝑅𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wex 1774  wcel 2099  Vcvv 3462   class class class wbr 5145  ccnv 5673  ccom 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5146  df-opab 5208  df-cnv 5682  df-co 5683
This theorem is referenced by:  codir  6124
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