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Theorem raaanv 4055
 Description: Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
Assertion
Ref Expression
raaanv (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem raaanv
StepHypRef Expression
1 nfv 1840 . 2 𝑦𝜑
2 nfv 1840 . 2 𝑥𝜓
31, 2raaan 4054 1 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384  ∀wral 2907 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-v 3188  df-dif 3558  df-nul 3892 This theorem is referenced by:  reusv3i  4835  f1mpt  6472  isclo2  20802  ntrk2imkb  37817
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