Theorem aaan 1580

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 1556	. . 3 ⊢ (∀𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑦𝜓))
3	2	albii 1463	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 ∧ ∀𝑦𝜓))
4		aaan.2	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	nfal 1569	. . 3 ⊢ Ⅎ𝑥∀𝑦𝜓
6	5	19.27 1554	. 2 ⊢ (∀𝑥(𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
7	3, 6	bitri 183	1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))

Syntax hints:   ∧ wa 103   ↔ wb 104  ∀wal 1346  Ⅎwnf 1453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-4 1503

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by: (None)