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Theorem raaan2 41699
Description: Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 4226. 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,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem raaan2
StepHypRef Expression
1 dfbi3 1036 . 2 ((𝐴 = ∅ ↔ 𝐵 = ∅) ↔ ((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
2 rzal 4217 . . . . 5 (𝐴 = ∅ → ∀𝑥𝐴𝑦𝐵 (𝜑𝜓))
32adantr 472 . . . 4 ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑥𝐴𝑦𝐵 (𝜑𝜓))
4 rzal 4217 . . . . 5 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
54adantr 472 . . . 4 ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑥𝐴 𝜑)
6 rzal 4217 . . . . 5 (𝐵 = ∅ → ∀𝑦𝐵 𝜓)
76adantl 473 . . . 4 ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑦𝐵 𝜓)
8 pm5.1 938 . . . 4 ((∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ∧ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
93, 5, 7, 8syl12anc 1475 . . 3 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
10 df-ne 2933 . . . . 5 (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
11 raaan2.1 . . . . . . 7 𝑦𝜑
1211r19.28z 4207 . . . . . 6 (𝐵 ≠ ∅ → (∀𝑦𝐵 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑦𝐵 𝜓)))
1312ralbidv 3124 . . . . 5 (𝐵 ≠ ∅ → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐵 𝜓)))
1410, 13sylbir 225 . . . 4 𝐵 = ∅ → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐵 𝜓)))
15 df-ne 2933 . . . . 5 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
16 nfcv 2902 . . . . . . 7 𝑥𝐵
17 raaan2.2 . . . . . . 7 𝑥𝜓
1816, 17nfral 3083 . . . . . 6 𝑥𝑦𝐵 𝜓
1918r19.27z 4214 . . . . 5 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐵 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
2015, 19sylbir 225 . . . 4 𝐴 = ∅ → (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐵 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
2114, 20sylan9bbr 739 . . 3 ((¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
229, 21jaoi 393 . 2 (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
231, 22sylbi 207 1 ((𝐴 = ∅ ↔ 𝐵 = ∅) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wnf 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:  2reu4a  41713
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