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Theorem raaan 4226
 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 4217 . . 3 (𝐴 = ∅ → ∀𝑥𝐴𝑦𝐴 (𝜑𝜓))
2 rzal 4217 . . 3 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
3 rzal 4217 . . 3 (𝐴 = ∅ → ∀𝑦𝐴 𝜓)
4 pm5.1 938 . . 3 ((∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ∧ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)) → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
51, 2, 3, 4syl12anc 1475 . 2 (𝐴 = ∅ → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
6 raaan.1 . . . . 5 𝑦𝜑
76r19.28z 4207 . . . 4 (𝐴 ≠ ∅ → (∀𝑦𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑦𝐴 𝜓)))
87ralbidv 3124 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓)))
9 nfcv 2902 . . . . 5 𝑥𝐴
10 raaan.2 . . . . 5 𝑥𝜓
119, 10nfral 3083 . . . 4 𝑥𝑦𝐴 𝜓
1211r19.27z 4214 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
138, 12bitrd 268 . 2 (𝐴 ≠ ∅ → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
145, 13pm2.61ine 3015 1 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383   = wceq 1632  Ⅎwnf 1857   ≠ wne 2932  ∀wral 3050  ∅c0 4058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-v 3342  df-dif 3718  df-nul 4059 This theorem is referenced by:  raaanv  4227
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