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Theorem reximdvva 3280
 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 470 . . 3 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
32reximdva 3277 . 2 ((𝜑𝑥𝐴) → (∃𝑦𝐵 𝜓 → ∃𝑦𝐵 𝜒))
43reximdva 3277 1 (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 → ∃𝑥𝐴𝑦𝐵 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2113  ∃wrex 3142 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-ral 3146  df-rex 3147 This theorem is referenced by:  reuop  6147  lcmgcdlem  15953  lsmelval2  19860  cpmadugsum  21489  axpasch  26730  frgrwopreglem5  28103  frgrwopreglem5ALT  28104  eulerpartlemgvv  31638  cusgr3cyclex  32387  cvmlift2lem10  32563  ftc1anclem6  34976  prprelprb  43686  reupr  43691
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