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Theorem exanres3 37800
Description: Equivalent expressions with restricted existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.)
Assertion
Ref Expression
exanres3 ((𝐵𝑉𝐶𝑊) → (∃𝑢𝐴 (𝐵 ∈ [𝑢]𝑅𝐶 ∈ [𝑢]𝑆) ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑆𝐶)))
Distinct variable groups:   𝑢,𝐵   𝑢,𝐶   𝑢,𝑉   𝑢,𝑊
Allowed substitution hints:   𝐴(𝑢)   𝑅(𝑢)   𝑆(𝑢)

Proof of Theorem exanres3
StepHypRef Expression
1 elecALTV 37770 . . . 4 ((𝑢 ∈ V ∧ 𝐵𝑉) → (𝐵 ∈ [𝑢]𝑅𝑢𝑅𝐵))
21el2v1 37723 . . 3 (𝐵𝑉 → (𝐵 ∈ [𝑢]𝑅𝑢𝑅𝐵))
3 elecALTV 37770 . . . 4 ((𝑢 ∈ V ∧ 𝐶𝑊) → (𝐶 ∈ [𝑢]𝑆𝑢𝑆𝐶))
43el2v1 37723 . . 3 (𝐶𝑊 → (𝐶 ∈ [𝑢]𝑆𝑢𝑆𝐶))
52, 4bi2anan9 636 . 2 ((𝐵𝑉𝐶𝑊) → ((𝐵 ∈ [𝑢]𝑅𝐶 ∈ [𝑢]𝑆) ↔ (𝑢𝑅𝐵𝑢𝑆𝐶)))
65rexbidv 3176 1 ((𝐵𝑉𝐶𝑊) → (∃𝑢𝐴 (𝐵 ∈ [𝑢]𝑅𝐶 ∈ [𝑢]𝑆) ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑆𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2098  wrex 3067  Vcvv 3473   class class class wbr 5152  [cec 8729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ec 8733
This theorem is referenced by:  exanres2  37801  br1cossres2  37944
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