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Theorem raaanv 4223
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 1995 . 2 𝑦𝜑
2 nfv 1995 . 2 𝑥𝜓
31, 2raaan 4222 1 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-v 3353  df-dif 3727  df-nul 4065
This theorem is referenced by:  reusv3i  5005  f1mpt  6662  isclo2  21114  ntrk2imkb  38862
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