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Theorem raaan 4421
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 4414 . . 3 (𝐴 = ∅ → ∀𝑥𝐴𝑦𝐴 (𝜑𝜓))
2 rzal 4414 . . 3 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
3 rzal 4414 . . 3 (𝐴 = ∅ → ∀𝑦𝐴 𝜓)
4 pm5.1 822 . . 3 ((∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ∧ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)) → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
51, 2, 3, 4syl12anc 835 . 2 (𝐴 = ∅ → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
6 raaan.1 . . . . 5 𝑦𝜑
76r19.28z 4404 . . . 4 (𝐴 ≠ ∅ → (∀𝑦𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑦𝐴 𝜓)))
87ralbidv 3165 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓)))
9 nfcv 2958 . . . . 5 𝑥𝐴
10 raaan.2 . . . . 5 𝑥𝜓
119, 10nfralw 3192 . . . 4 𝑥𝑦𝐴 𝜓
1211r19.27z 4411 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
138, 12bitrd 282 . 2 (𝐴 ≠ ∅ → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
145, 13pm2.61ine 3073 1 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wnf 1785  wne 2990  wral 3109  c0 4246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-dif 3887  df-nul 4247
This theorem is referenced by: (None)
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