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Theorem ralrimivv 2571
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 258 . . 3 (𝜑 → (𝑥𝐴 → (𝑦𝐵𝜓)))
32ralrimdv 2569 . 2 (𝜑 → (𝑥𝐴 → ∀𝑦𝐵 𝜓))
43ralrimiv 2562 1 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2160  wral 2468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-4 1521  ax-17 1537
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-ral 2473
This theorem is referenced by:  ralrimivva  2572  ralrimdvv  2574  reuind  2957  ssrel2  4737  f1o2ndf1  6257  smoiso  6331  nndifsnid  6536  receuap  8661  lbreu  8937  0subm  12959  insubm  12960  iscmnd  13262  quscrng  13872  tgcl  14049  topbas  14052  epttop  14075  restbasg  14153  txbas  14243  txbasval  14252  blfps  14394  blf  14395  blbas  14418
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