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Theorem exan3 37163
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 37134 . . . 4 ((𝑢 ∈ V ∧ 𝐴𝑉) → (𝐴 ∈ [𝑢]𝑅𝑢𝑅𝐴))
21el2v1 37085 . . 3 (𝐴𝑉 → (𝐴 ∈ [𝑢]𝑅𝑢𝑅𝐴))
3 elecALTV 37134 . . . 4 ((𝑢 ∈ V ∧ 𝐵𝑊) → (𝐵 ∈ [𝑢]𝑅𝑢𝑅𝐵))
43el2v1 37085 . . 3 (𝐵𝑊 → (𝐵 ∈ [𝑢]𝑅𝑢𝑅𝐵))
52, 4bi2anan9 638 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∈ [𝑢]𝑅𝐵 ∈ [𝑢]𝑅) ↔ (𝑢𝑅𝐴𝑢𝑅𝐵)))
65exbidv 1925 1 ((𝐴𝑉𝐵𝑊) → (∃𝑢(𝐴 ∈ [𝑢]𝑅𝐵 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wex 1782  wcel 2107  Vcvv 3475   class class class wbr 5149  [cec 8701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ec 8705
This theorem is referenced by:  brcoss2  37302
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