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Theorem raaan2 4460
Description: Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 4456. 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 1041 . 2 ((𝐴 = ∅ ↔ 𝐵 = ∅) ↔ ((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
2 rzal 4449 . . . . 5 (𝐴 = ∅ → ∀𝑥𝐴𝑦𝐵 (𝜑𝜓))
32adantr 481 . . . 4 ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑥𝐴𝑦𝐵 (𝜑𝜓))
4 rzal 4449 . . . . 5 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
54adantr 481 . . . 4 ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑥𝐴 𝜑)
6 rzal 4449 . . . . 5 (𝐵 = ∅ → ∀𝑦𝐵 𝜓)
76adantl 482 . . . 4 ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑦𝐵 𝜓)
8 pm5.1 819 . . . 4 ((∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ∧ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
93, 5, 7, 8syl12anc 832 . . 3 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
10 df-ne 3014 . . . . 5 (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
11 raaan2.1 . . . . . . 7 𝑦𝜑
1211r19.28z 4439 . . . . . 6 (𝐵 ≠ ∅ → (∀𝑦𝐵 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑦𝐵 𝜓)))
1312ralbidv 3194 . . . . 5 (𝐵 ≠ ∅ → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐵 𝜓)))
1410, 13sylbir 236 . . . 4 𝐵 = ∅ → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐵 𝜓)))
15 df-ne 3014 . . . . 5 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
16 nfcv 2974 . . . . . . 7 𝑥𝐵
17 raaan2.2 . . . . . . 7 𝑥𝜓
1816, 17nfralw 3222 . . . . . 6 𝑥𝑦𝐵 𝜓
1918r19.27z 4446 . . . . 5 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐵 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
2015, 19sylbir 236 . . . 4 𝐴 = ∅ → (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐵 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
2114, 20sylan9bbr 511 . . 3 ((¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
229, 21jaoi 851 . 2 (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
231, 22sylbi 218 1 ((𝐴 = ∅ ↔ 𝐵 = ∅) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wnf 1775  wne 3013  wral 3135  c0 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-dif 3936  df-nul 4289
This theorem is referenced by:  2reu4lem  4461
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