Theorem raaan 3658

Description: Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)

Hypotheses

Ref	Expression
raaan.1	⊢ Ⅎyφ
raaan.2	⊢ Ⅎxψ

Assertion

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

Distinct variable group:   x,y,A

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

Proof of Theorem raaan

Step	Hyp	Ref	Expression
1		rzal 3652	. . 3 ⊢ (A = ∅ → ∀x ∈ A ∀y ∈ A (φ ∧ ψ))
2		rzal 3652	. . 3 ⊢ (A = ∅ → ∀x ∈ A φ)
3		rzal 3652	. . 3 ⊢ (A = ∅ → ∀y ∈ A ψ)
4		pm5.1 830	. . 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 1180	. 2 ⊢ (A = ∅ → (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ)))
6		raaan.1	. . . . 5 ⊢ Ⅎyφ
7	6	r19.28z 3643	. . . 4 ⊢ (A ≠ ∅ → (∀y ∈ A (φ ∧ ψ) ↔ (φ ∧ ∀y ∈ A ψ)))
8	7	ralbidv 2635	. . 3 ⊢ (A ≠ ∅ → (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ ∀x ∈ A (φ ∧ ∀y ∈ A ψ)))
9		nfcv 2490	. . . . 5 ⊢ ℲxA
10		raaan.2	. . . . 5 ⊢ Ⅎxψ
11	9, 10	nfral 2668	. . . 4 ⊢ Ⅎx∀y ∈ A ψ
12	11	r19.27z 3649	. . 3 ⊢ (A ≠ ∅ → (∀x ∈ A (φ ∧ ∀y ∈ A ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ)))
13	8, 12	bitrd 244	. 2 ⊢ (A ≠ ∅ → (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ)))
14	5, 13	pm2.61ine 2593	1 ⊢ (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ))

Syntax hints:   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642   ≠ wne 2517  ∀wral 2615  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by: (None)