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Theorem exanres 36053
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 5832 . . . . 5 (𝐵𝑉 → (𝑢(𝑅𝐴)𝐵 ↔ (𝑢𝐴𝑢𝑅𝐵)))
2 brres 5832 . . . . 5 (𝐶𝑊 → (𝑢(𝑆𝐴)𝐶 ↔ (𝑢𝐴𝑢𝑆𝐶)))
31, 2bi2anan9 639 . . . 4 ((𝐵𝑉𝐶𝑊) → ((𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ ((𝑢𝐴𝑢𝑅𝐵) ∧ (𝑢𝐴𝑢𝑆𝐶))))
4 anandi 676 . . . 4 ((𝑢𝐴 ∧ (𝑢𝑅𝐵𝑢𝑆𝐶)) ↔ ((𝑢𝐴𝑢𝑅𝐵) ∧ (𝑢𝐴𝑢𝑆𝐶)))
53, 4bitr4di 292 . . 3 ((𝐵𝑉𝐶𝑊) → ((𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ (𝑢𝐴 ∧ (𝑢𝑅𝐵𝑢𝑆𝐶))))
65exbidv 1928 . 2 ((𝐵𝑉𝐶𝑊) → (∃𝑢(𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ ∃𝑢(𝑢𝐴 ∧ (𝑢𝑅𝐵𝑢𝑆𝐶))))
7 df-rex 3059 . 2 (∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑆𝐶) ↔ ∃𝑢(𝑢𝐴 ∧ (𝑢𝑅𝐵𝑢𝑆𝐶)))
86, 7bitr4di 292 1 ((𝐵𝑉𝐶𝑊) → (∃𝑢(𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑆𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1786  wcel 2114  wrex 3054   class class class wbr 5030  cres 5527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-res 5537
This theorem is referenced by:  exanres2  36055  br1cossres  36185
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