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Theorem raaan 4032
Description: Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)
Hypotheses
Ref Expression
raaan.1 𝑦𝜑
raaan.2 𝑥𝜓
Assertion
Ref Expression
raaan (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem raaan
StepHypRef Expression
1 rzal 4025 . . 3 (𝐴 = ∅ → ∀𝑥𝐴𝑦𝐴 (𝜑𝜓))
2 rzal 4025 . . 3 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
3 rzal 4025 . . 3 (𝐴 = ∅ → ∀𝑦𝐴 𝜓)
4 pm5.1 898 . . 3 ((∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ∧ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)) → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
51, 2, 3, 4syl12anc 1316 . 2 (𝐴 = ∅ → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
6 raaan.1 . . . . 5 𝑦𝜑
76r19.28z 4015 . . . 4 (𝐴 ≠ ∅ → (∀𝑦𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑦𝐴 𝜓)))
87ralbidv 2969 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓)))
9 nfcv 2751 . . . . 5 𝑥𝐴
10 raaan.2 . . . . 5 𝑥𝜓
119, 10nfral 2929 . . . 4 𝑥𝑦𝐴 𝜓
1211r19.27z 4022 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
138, 12bitrd 267 . 2 (𝐴 ≠ ∅ → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
145, 13pm2.61ine 2865 1 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  wnf 1699  wne 2780  wral 2896  c0 3874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-v 3175  df-dif 3543  df-nul 3875
This theorem is referenced by:  raaanv  4033
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