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Theorem reximdvva 3228
Description: Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by AV, 5-Jan-2022.)
Hypothesis
Ref Expression
reximdvva.1 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
reximdvva (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 → ∃𝑥𝐴𝑦𝐵 𝜒))
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝜑
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem reximdvva
StepHypRef Expression
1 reximdvva.1 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
21anassrs 461 . . 3 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
32reximdva 3225 . 2 ((𝜑𝑥𝐴) → (∃𝑦𝐵 𝜓 → ∃𝑦𝐵 𝜒))
43reximdva 3225 1 (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 → ∃𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wcel 2164  wrex 3118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-ral 3122  df-rex 3123
This theorem is referenced by:  lcmgcdlem  15699  lsmelval2  19451  cpmadugsum  21060  axpasch  26247  frgrwopreglem5  27698  frgrwopreglem5ALT  27699  eulerpartlemgvv  30979  cvmlift2lem10  31836  ftc1anclem6  34032
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