Theorem raaanv 4465

Description: Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)

Assertion

Ref	Expression
raaanv	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem raaanv

Step	Hyp	Ref	Expression
1		rzal 4456	. . 3 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓))
2		rzal 4456	. . 3 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
3		rzal 4456	. . 3 ⊢ (𝐴 = ∅ → ∀𝑦 ∈ 𝐴 𝜓)
4		pm5.1 823	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
5	1, 2, 3, 4	syl12anc 836	. 2 ⊢ (𝐴 = ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
6		r19.28zv 4448	. . . 4 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
7	6	ralbidv 3155	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
8		r19.27zv 4453	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
9	7, 8	bitrd 279	. 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
10	5, 9	pm2.61ine 3011	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ≠ wne 2928  ∀wral 3047  ∅c0 4280

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-9 2121  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-ne 2929  df-ral 3048  df-dif 3900  df-nul 4281

This theorem is referenced by:  reusv3i  5340  f1mpt  7195  isclo2  23003  ntrk2imkb  44129