Theorem raaan2 4473

Description: Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 4469. 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 1049	. 2 ⊢ ((𝐴 = ∅ ↔ 𝐵 = ∅) ↔ ((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
2		rzal 4445	. . . . 5 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓))
3	2	adantr 480	. . . 4 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓))
4		rzal 4445	. . . . 5 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
5	4	adantr 480	. . . 4 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑥 ∈ 𝐴 𝜑)
6		rzal 4445	. . . . 5 ⊢ (𝐵 = ∅ → ∀𝑦 ∈ 𝐵 𝜓)
7	6	adantl 481	. . . 4 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∀𝑦 ∈ 𝐵 𝜓)
8		pm5.1 823	. . . 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 836	. . 3 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
10		df-ne 2931	. . . . 5 ⊢ (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
11		raaan2.1	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
12	11	r19.28z 4453	. . . . . 6 ⊢ (𝐵 ≠ ∅ → (∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
13	12	ralbidv 3157	. . . . 5 ⊢ (𝐵 ≠ ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
14	10, 13	sylbir 235	. . . 4 ⊢ (¬ 𝐵 = ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
15		df-ne 2931	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
16		nfcv 2896	. . . . . . 7 ⊢ Ⅎ𝑥𝐵
17		raaan2.2	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
18	16, 17	nfralw 3281	. . . . . 6 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐵 𝜓
19	18	r19.27z 4461	. . . . 5 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
20	15, 19	sylbir 235	. . . 4 ⊢ (¬ 𝐴 = ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
21	14, 20	sylan9bbr 510	. . 3 ⊢ ((¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
22	9, 21	jaoi 857	. 2 ⊢ (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
23	1, 22	sylbi 217	1 ⊢ ((𝐴 = ∅ ↔ 𝐵 = ∅) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541  Ⅎwnf 1784   ≠ wne 2930  ∀wral 3049  ∅c0 4283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-dif 3902  df-nul 4284

This theorem is referenced by:  2reu4lem  4474