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Theorem exanres3 35671
 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 35645 . . . 4 ((𝑢 ∈ V ∧ 𝐵𝑉) → (𝐵 ∈ [𝑢]𝑅𝑢𝑅𝐵))
21el2v1 35608 . . 3 (𝐵𝑉 → (𝐵 ∈ [𝑢]𝑅𝑢𝑅𝐵))
3 elecALTV 35645 . . . 4 ((𝑢 ∈ V ∧ 𝐶𝑊) → (𝐶 ∈ [𝑢]𝑆𝑢𝑆𝐶))
43el2v1 35608 . . 3 (𝐶𝑊 → (𝐶 ∈ [𝑢]𝑆𝑢𝑆𝐶))
52, 4bi2anan9 638 . 2 ((𝐵𝑉𝐶𝑊) → ((𝐵 ∈ [𝑢]𝑅𝐶 ∈ [𝑢]𝑆) ↔ (𝑢𝑅𝐵𝑢𝑆𝐶)))
65rexbidv 3283 1 ((𝐵𝑉𝐶𝑊) → (∃𝑢𝐴 (𝐵 ∈ [𝑢]𝑅𝐶 ∈ [𝑢]𝑆) ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑆𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2114  ∃wrex 3131  Vcvv 3469   class class class wbr 5042  [cec 8274 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-cnv 5540  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-ec 8278 This theorem is referenced by:  exanres2  35672  br1cossres2  35803
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