Theorem aaan 1519

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 1495	. . 3 ⊢ (∀𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑦𝜓))
3	2	albii 1399	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 ∧ ∀𝑦𝜓))
4		aaan.2	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	nfal 1508	. . 3 ⊢ Ⅎ𝑥∀𝑦𝜓
6	5	19.27 1493	. 2 ⊢ (∀𝑥(𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
7	3, 6	bitri 182	1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))

Syntax hints:   ∧ wa 102   ↔ wb 103  ∀wal 1282  Ⅎwnf 1389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-4 1440

This theorem depends on definitions:  df-bi 115  df-nf 1390

This theorem is referenced by: (None)