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Theorem raaan2 4525
Description: Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 4521. It is necessary that either both restricting classes are empty or both are not empty. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
Hypotheses
Ref Expression
raaan2.1 𝑦𝜑
raaan2.2 𝑥𝜓
Assertion
Ref Expression
raaan2 ((𝐴 = ∅ ↔ 𝐵 = ∅) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑦)

Proof of Theorem raaan2
StepHypRef Expression
1 dfbi3 1046 . 2 ((𝐴 = ∅ ↔ 𝐵 = ∅) ↔ ((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
2 rzal 4509 . . . . 5 (𝐴 = ∅ → ∀𝑥𝐴𝑦𝐵 (𝜑𝜓))
32adantr 479 . . . 4 ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑥𝐴𝑦𝐵 (𝜑𝜓))
4 rzal 4509 . . . . 5 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
54adantr 479 . . . 4 ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑥𝐴 𝜑)
6 rzal 4509 . . . . 5 (𝐵 = ∅ → ∀𝑦𝐵 𝜓)
76adantl 480 . . . 4 ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑦𝐵 𝜓)
8 pm5.1 820 . . . 4 ((∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ∧ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
93, 5, 7, 8syl12anc 833 . . 3 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
10 df-ne 2939 . . . . 5 (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
11 raaan2.1 . . . . . . 7 𝑦𝜑
1211r19.28z 4498 . . . . . 6 (𝐵 ≠ ∅ → (∀𝑦𝐵 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑦𝐵 𝜓)))
1312ralbidv 3175 . . . . 5 (𝐵 ≠ ∅ → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐵 𝜓)))
1410, 13sylbir 234 . . . 4 𝐵 = ∅ → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐵 𝜓)))
15 df-ne 2939 . . . . 5 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
16 nfcv 2901 . . . . . . 7 𝑥𝐵
17 raaan2.2 . . . . . . 7 𝑥𝜓
1816, 17nfralw 3306 . . . . . 6 𝑥𝑦𝐵 𝜓
1918r19.27z 4505 . . . . 5 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐵 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
2015, 19sylbir 234 . . . 4 𝐴 = ∅ → (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐵 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
2114, 20sylan9bbr 509 . . 3 ((¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
229, 21jaoi 853 . 2 (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
231, 22sylbi 216 1 ((𝐴 = ∅ ↔ 𝐵 = ∅) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 843   = wceq 1539  wnf 1783  wne 2938  wral 3059  c0 4323
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-dif 3952  df-nul 4324
This theorem is referenced by:  2reu4lem  4526
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