Theorem aaan 1884

Description: Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)

Hypotheses

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

Assertion

Ref	Expression
aaan	⊢ (∀x∀y(φ ∧ ψ) ↔ (∀xφ ∧ ∀yψ))

Proof of Theorem aaan

Step	Hyp	Ref	Expression
1		aaan.1	. . . 4 ⊢ Ⅎyφ
2	1	19.28 1870	. . 3 ⊢ (∀y(φ ∧ ψ) ↔ (φ ∧ ∀yψ))
3	2	albii 1566	. 2 ⊢ (∀x∀y(φ ∧ ψ) ↔ ∀x(φ ∧ ∀yψ))
4		aaan.2	. . . 4 ⊢ Ⅎxψ
5	4	nfal 1842	. . 3 ⊢ Ⅎx∀yψ
6	5	19.27 1869	. 2 ⊢ (∀x(φ ∧ ∀yψ) ↔ (∀xφ ∧ ∀yψ))
7	3, 6	bitri 240	1 ⊢ (∀x∀y(φ ∧ ψ) ↔ (∀xφ ∧ ∀yψ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544

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-7 1734  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by:  mo  2226  2mo  2282  2eu4  2287