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Theorem exan3 36041
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 36017 . . . 4 ((𝑢 ∈ V ∧ 𝐴𝑉) → (𝐴 ∈ [𝑢]𝑅𝑢𝑅𝐴))
21el2v1 35982 . . 3 (𝐴𝑉 → (𝐴 ∈ [𝑢]𝑅𝑢𝑅𝐴))
3 elecALTV 36017 . . . 4 ((𝑢 ∈ V ∧ 𝐵𝑊) → (𝐵 ∈ [𝑢]𝑅𝑢𝑅𝐵))
43el2v1 35982 . . 3 (𝐵𝑊 → (𝐵 ∈ [𝑢]𝑅𝑢𝑅𝐵))
52, 4bi2anan9 639 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∈ [𝑢]𝑅𝐵 ∈ [𝑢]𝑅) ↔ (𝑢𝑅𝐴𝑢𝑅𝐵)))
65exbidv 1927 1 ((𝐴𝑉𝐵𝑊) → (∃𝑢(𝐴 ∈ [𝑢]𝑅𝐵 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1786  wcel 2113  Vcvv 3397   class class class wbr 5027  [cec 8311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-br 5028  df-opab 5090  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ec 8315
This theorem is referenced by:  brcoss2  36167
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