Theorem raaanv 4474

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 4449	. . 3 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓))
2		rzal 4449	. . 3 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
3		rzal 4449	. . 3 ⊢ (𝐴 = ∅ → ∀𝑦 ∈ 𝐴 𝜓)
4		pm5.1 836	. . 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 850	. 2 ⊢ (𝐴 = ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
6		r19.28zv 4461	. . . 4 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
7	6	ralbidv 3185	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
8		r19.27zv 4466	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
9	7, 8	bitrd 282	. 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
10	5, 9	pm2.61ine 3038	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ≠ wne 2955  ∀wral 3076  ∅c0 4278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-ne 2956  df-ral 3077  df-dif 4047  df-nul 4279

This theorem is used by:  reusv3i  5365  f1mpt  7265  isclo2  23406  ntrk2imkb  45037