Theorem raaanv 3658

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 3651	. . 3 ⊢ (A = ∅ → ∀x ∈ A ∀y ∈ A (φ ∧ ψ))
2		rzal 3651	. . 3 ⊢ (A = ∅ → ∀x ∈ A φ)
3		rzal 3651	. . 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		r19.28zv 3645	. . . 4 ⊢ (A ≠ ∅ → (∀y ∈ A (φ ∧ ψ) ↔ (φ ∧ ∀y ∈ A ψ)))
7	6	ralbidv 2634	. . 3 ⊢ (A ≠ ∅ → (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ ∀x ∈ A (φ ∧ ∀y ∈ A ψ)))
8		r19.27zv 3649	. . 3 ⊢ (A ≠ ∅ → (∀x ∈ A (φ ∧ ∀y ∈ A ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ)))
9	7, 8	bitrd 244	. 2 ⊢ (A ≠ ∅ → (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ)))
10	5, 9	pm2.61ine 2592	1 ⊢ (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ))

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ≠ wne 2516  ∀wral 2614  ∅c0 3550

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 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551

This theorem is referenced by: (None)