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Theorem ralrimdvv 2453
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005.)
Hypothesis
Ref Expression
ralrimdvv.1 (𝜑 → (𝜓 → ((𝑥𝐴𝑦𝐵) → 𝜒)))
Assertion
Ref Expression
ralrimdvv (𝜑 → (𝜓 → ∀𝑥𝐴𝑦𝐵 𝜒))
Distinct variable groups:   𝑥,𝑦,𝜑   𝜓,𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜒(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem ralrimdvv
StepHypRef Expression
1 ralrimdvv.1 . . . 4 (𝜑 → (𝜓 → ((𝑥𝐴𝑦𝐵) → 𝜒)))
21imp 122 . . 3 ((𝜑𝜓) → ((𝑥𝐴𝑦𝐵) → 𝜒))
32ralrimivv 2450 . 2 ((𝜑𝜓) → ∀𝑥𝐴𝑦𝐵 𝜒)
43ex 113 1 (𝜑 → (𝜓 → ∀𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1436  wral 2355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-4 1443  ax-17 1462
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-ral 2360
This theorem is referenced by:  ralrimdvva  2454
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