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Theorem reximdvva 3157
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 683 . . 3 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
32reximdva 3155 . 2 ((𝜑𝑥𝐴) → (∃𝑦𝐵 𝜓 → ∃𝑦𝐵 𝜒))
43reximdva 3155 1 (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 → ∃𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  wrex 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-ral 3055  df-rex 3056
This theorem is referenced by:  lcmgcdlem  15521  lsmelval2  19287  cpmadugsum  20885  axpasch  26020  frgrwopreglem5  27475  frgrwopreglem5ALT  27476  eulerpartlemgvv  30747  cvmlift2lem10  31601  ftc1anclem6  33803
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