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Theorem exanres 35084
 Description: Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 2-May-2021.)
Assertion
Ref Expression
exanres ((𝐵𝑉𝐶𝑊) → (∃𝑢(𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ ∃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 exanres
StepHypRef Expression
1 brres 5741 . . . . 5 (𝐵𝑉 → (𝑢(𝑅𝐴)𝐵 ↔ (𝑢𝐴𝑢𝑅𝐵)))
2 brres 5741 . . . . 5 (𝐶𝑊 → (𝑢(𝑆𝐴)𝐶 ↔ (𝑢𝐴𝑢𝑆𝐶)))
31, 2bi2anan9 635 . . . 4 ((𝐵𝑉𝐶𝑊) → ((𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ ((𝑢𝐴𝑢𝑅𝐵) ∧ (𝑢𝐴𝑢𝑆𝐶))))
4 anandi 672 . . . 4 ((𝑢𝐴 ∧ (𝑢𝑅𝐵𝑢𝑆𝐶)) ↔ ((𝑢𝐴𝑢𝑅𝐵) ∧ (𝑢𝐴𝑢𝑆𝐶)))
53, 4syl6bbr 290 . . 3 ((𝐵𝑉𝐶𝑊) → ((𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ (𝑢𝐴 ∧ (𝑢𝑅𝐵𝑢𝑆𝐶))))
65exbidv 1899 . 2 ((𝐵𝑉𝐶𝑊) → (∃𝑢(𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ ∃𝑢(𝑢𝐴 ∧ (𝑢𝑅𝐵𝑢𝑆𝐶))))
7 df-rex 3111 . 2 (∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑆𝐶) ↔ ∃𝑢(𝑢𝐴 ∧ (𝑢𝑅𝐵𝑢𝑆𝐶)))
86, 7syl6bbr 290 1 ((𝐵𝑉𝐶𝑊) → (∃𝑢(𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑆𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∃wex 1761   ∈ wcel 2081  ∃wrex 3106   class class class wbr 4962   ↾ cres 5445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-xp 5449  df-res 5455 This theorem is referenced by:  exanres2  35086  br1cossres  35215
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