Theorem raaan2 4478

Description: Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 4474. 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	⊢ Ⅎ𝑦𝜑
raaan2.2	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
raaan2	⊢ ((𝐴 = ∅ ↔ 𝐵 = ∅) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 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

Step	Hyp	Ref	Expression
1		dfbi3 1061	. 2 ⊢ ((𝐴 = ∅ ↔ 𝐵 = ∅) ↔ ((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
2		rzal 4450	. . . . 5 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓))
3	2	adantr 484	. . . 4 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓))
4		rzal 4450	. . . . 5 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
5	4	adantr 484	. . . 4 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑥 ∈ 𝐴 𝜑)
6		rzal 4450	. . . . 5 ⊢ (𝐵 = ∅ → ∀𝑦 ∈ 𝐵 𝜓)
7	6	adantl 485	. . . 4 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑦 ∈ 𝐵 𝜓)
8		pm5.1 833	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
9	3, 5, 7, 8	syl12anc 847	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
10		df-ne 2960	. . . . 5 ⊢ (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
11		raaan2.1	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
12	11	r19.28z 4458	. . . . . 6 ⊢ (𝐵 ≠ ∅ → (∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
13	12	ralbidv 3187	. . . . 5 ⊢ (𝐵 ≠ ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
14	10, 13	sylbir 237	. . . 4 ⊢ (¬ 𝐵 = ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
15		df-ne 2960	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
16		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑥𝐵
17		raaan2.2	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
18	16, 17	nfralw 3311	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐵 𝜓
19	18	r19.27z 4466	. . . . 5 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
20	15, 19	sylbir 237	. . . 4 ⊢ (¬ 𝐴 = ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
21	14, 20	sylan9bbr 518	. . 3 ⊢ ((¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
22	9, 21	jaoi 868	. 2 ⊢ (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
23	1, 22	sylbi 219	1 ⊢ ((𝐴 = ∅ ↔ 𝐵 = ∅) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1562  Ⅎwnf 1805   ≠ wne 2959  ∀wral 3078  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-dif 3909  df-nul 4288

This theorem is referenced by:  2reu4lem  4479