Theorem aaan 1640

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 1616	. . 3 ⊢ (∀𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑦𝜓))
3	2	albii 1523	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 ∧ ∀𝑦𝜓))
4		aaan.2	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	nfal 1629	. . 3 ⊢ Ⅎ𝑥∀𝑦𝜓
6	5	19.27 1614	. 2 ⊢ (∀𝑥(𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
7	3, 6	bitri 184	1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105  ∀wal 1400  Ⅎwnf 1513

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-4 1563

This proof depends on definitions:  df-bi 117  df-nf 1514

This theorem is used by: (None)