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Theorem raaanv 4464
 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 rzal 4456 . . 3 (𝐴 = ∅ → ∀𝑥𝐴𝑦𝐴 (𝜑𝜓))
2 rzal 4456 . . 3 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
3 rzal 4456 . . 3 (𝐴 = ∅ → ∀𝑦𝐴 𝜓)
4 pm5.1 821 . . 3 ((∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ∧ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)) → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
51, 2, 3, 4syl12anc 834 . 2 (𝐴 = ∅ → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
6 r19.28zv 4449 . . . 4 (𝐴 ≠ ∅ → (∀𝑦𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑦𝐴 𝜓)))
76ralbidv 3200 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓)))
8 r19.27zv 4454 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
97, 8bitrd 281 . 2 (𝐴 ≠ ∅ → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
105, 9pm2.61ine 3103 1 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1536   ≠ wne 3019  ∀wral 3141  ∅c0 4294 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-11 2160  ax-12 2176  ax-ext 2796 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-dif 3942  df-nul 4295 This theorem is referenced by:  reusv3i  5308  f1mpt  7022  isclo2  21699  ntrk2imkb  40393
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