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Theorem exan3 38296
Description: Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.)
Assertion
Ref Expression
exan3 ((𝐴𝑉𝐵𝑊) → (∃𝑢(𝐴 ∈ [𝑢]𝑅𝐵 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝑉   𝑢,𝑊
Allowed substitution hint:   𝑅(𝑢)

Proof of Theorem exan3
StepHypRef Expression
1 elecALTV 38268 . . . 4 ((𝑢 ∈ V ∧ 𝐴𝑉) → (𝐴 ∈ [𝑢]𝑅𝑢𝑅𝐴))
21el2v1 38225 . . 3 (𝐴𝑉 → (𝐴 ∈ [𝑢]𝑅𝑢𝑅𝐴))
3 elecALTV 38268 . . . 4 ((𝑢 ∈ V ∧ 𝐵𝑊) → (𝐵 ∈ [𝑢]𝑅𝑢𝑅𝐵))
43el2v1 38225 . . 3 (𝐵𝑊 → (𝐵 ∈ [𝑢]𝑅𝑢𝑅𝐵))
52, 4bi2anan9 638 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∈ [𝑢]𝑅𝐵 ∈ [𝑢]𝑅) ↔ (𝑢𝑅𝐴𝑢𝑅𝐵)))
65exbidv 1920 1 ((𝐴𝑉𝐵𝑊) → (∃𝑢(𝐴 ∈ [𝑢]𝑅𝐵 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wex 1778  wcel 2107  Vcvv 3479   class class class wbr 5142  [cec 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ec 8748
This theorem is referenced by:  brcoss2  38434
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