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Theorem ralrimivv 2417
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004.)
Hypothesis
Ref Expression
ralrimivv.1 (𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜓))
Assertion
Ref Expression
ralrimivv (𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)
Distinct variable groups:   𝑥,𝑦,𝜑   𝑦,𝐴
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem ralrimivv
StepHypRef Expression
1 ralrimivv.1 . . . 4 (𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜓))
21expd 249 . . 3 (𝜑 → (𝑥𝐴 → (𝑦𝐵𝜓)))
32ralrimdv 2415 . 2 (𝜑 → (𝑥𝐴 → ∀𝑦𝐵 𝜓))
43ralrimiv 2408 1 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416  ax-17 1435
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-ral 2328
This theorem is referenced by:  ralrimivva  2418  ralrimdvv  2420  reuind  2767  ssrel2  4458  f1o2ndf1  5877  smoiso  5948  receuap  7724
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