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Theorem exanres3 35434
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 35408 . . . 4 ((𝑢 ∈ V ∧ 𝐵𝑉) → (𝐵 ∈ [𝑢]𝑅𝑢𝑅𝐵))
21el2v1 35371 . . 3 (𝐵𝑉 → (𝐵 ∈ [𝑢]𝑅𝑢𝑅𝐵))
3 elecALTV 35408 . . . 4 ((𝑢 ∈ V ∧ 𝐶𝑊) → (𝐶 ∈ [𝑢]𝑆𝑢𝑆𝐶))
43el2v1 35371 . . 3 (𝐶𝑊 → (𝐶 ∈ [𝑢]𝑆𝑢𝑆𝐶))
52, 4bi2anan9 635 . 2 ((𝐵𝑉𝐶𝑊) → ((𝐵 ∈ [𝑢]𝑅𝐶 ∈ [𝑢]𝑆) ↔ (𝑢𝑅𝐵𝑢𝑆𝐶)))
65rexbidv 3294 1 ((𝐵𝑉𝐶𝑊) → (∃𝑢𝐴 (𝐵 ∈ [𝑢]𝑅𝐶 ∈ [𝑢]𝑆) ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑆𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  wrex 3136  Vcvv 3492   class class class wbr 5057  [cec 8276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ec 8280
This theorem is referenced by:  exanres2  35435  br1cossres2  35565
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