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Theorem exanres 36430
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 5898 . . . . 5 (𝐵𝑉 → (𝑢(𝑅𝐴)𝐵 ↔ (𝑢𝐴𝑢𝑅𝐵)))
2 brres 5898 . . . . 5 (𝐶𝑊 → (𝑢(𝑆𝐴)𝐶 ↔ (𝑢𝐴𝑢𝑆𝐶)))
31, 2bi2anan9 636 . . . 4 ((𝐵𝑉𝐶𝑊) → ((𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ ((𝑢𝐴𝑢𝑅𝐵) ∧ (𝑢𝐴𝑢𝑆𝐶))))
4 anandi 673 . . . 4 ((𝑢𝐴 ∧ (𝑢𝑅𝐵𝑢𝑆𝐶)) ↔ ((𝑢𝐴𝑢𝑅𝐵) ∧ (𝑢𝐴𝑢𝑆𝐶)))
53, 4bitr4di 289 . . 3 ((𝐵𝑉𝐶𝑊) → ((𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ (𝑢𝐴 ∧ (𝑢𝑅𝐵𝑢𝑆𝐶))))
65exbidv 1924 . 2 ((𝐵𝑉𝐶𝑊) → (∃𝑢(𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ ∃𝑢(𝑢𝐴 ∧ (𝑢𝑅𝐵𝑢𝑆𝐶))))
7 df-rex 3070 . 2 (∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑆𝐶) ↔ ∃𝑢(𝑢𝐴 ∧ (𝑢𝑅𝐵𝑢𝑆𝐶)))
86, 7bitr4di 289 1 ((𝐵𝑉𝐶𝑊) → (∃𝑢(𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑆𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wex 1782  wcel 2106  wrex 3065   class class class wbr 5074  cres 5591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-res 5601
This theorem is referenced by:  exanres2  36432  br1cossres  36562
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