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Theorem raaan 4269
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 4262 . . 3 (𝐴 = ∅ → ∀𝑥𝐴𝑦𝐴 (𝜑𝜓))
2 rzal 4262 . . 3 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
3 rzal 4262 . . 3 (𝐴 = ∅ → ∀𝑦𝐴 𝜓)
4 pm5.1 845 . . 3 ((∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ∧ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)) → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
51, 2, 3, 4syl12anc 856 . 2 (𝐴 = ∅ → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
6 raaan.1 . . . . 5 𝑦𝜑
76r19.28z 4252 . . . 4 (𝐴 ≠ ∅ → (∀𝑦𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑦𝐴 𝜓)))
87ralbidv 3170 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓)))
9 nfcv 2944 . . . . 5 𝑥𝐴
10 raaan.2 . . . . 5 𝑥𝜓
119, 10nfral 3129 . . . 4 𝑥𝑦𝐴 𝜓
1211r19.27z 4259 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
138, 12bitrd 270 . 2 (𝐴 ≠ ∅ → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
145, 13pm2.61ine 3057 1 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1637  wnf 1863  wne 2974  wral 3092  c0 4110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-v 3389  df-dif 3766  df-nul 4111
This theorem is referenced by:  raaanv  4270
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