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Theorem raaanv 3328
 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 1421 . 2 𝑦𝜑
2 nfv 1421 . 2 𝑥𝜓
31, 2raaan 3327 1 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∀wral 2306 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311 This theorem is referenced by:  reusv3i  4191  f1mpt  5410
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