Theorem aaan 2328

Description: Distribute universal quantifiers. (Contributed by NM, 12-Aug-1993.) Avoid ax-10 2137. (Revised by Gino Giotto, 21-Nov-2024.)

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		19.26-2 1874	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓))
2		aaan.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
3	2	19.3 2195	. . . 4 ⊢ (∀𝑦𝜑 ↔ 𝜑)
4	3	albii 1822	. . 3 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑥𝜑)
5		alcom 2156	. . . 4 ⊢ (∀𝑥∀𝑦𝜓 ↔ ∀𝑦∀𝑥𝜓)
6		aaan.2	. . . . . 6 ⊢ Ⅎ𝑥𝜓
7	6	19.3 2195	. . . . 5 ⊢ (∀𝑥𝜓 ↔ 𝜓)
8	7	albii 1822	. . . 4 ⊢ (∀𝑦∀𝑥𝜓 ↔ ∀𝑦𝜓)
9	5, 8	bitri 274	. . 3 ⊢ (∀𝑥∀𝑦𝜓 ↔ ∀𝑦𝜓)
10	4, 9	anbi12i 627	. 2 ⊢ ((∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
11	1, 10	bitri 274	1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))

Syntax hints:   ↔ wb 205   ∧ wa 396  ∀wal 1537  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-nf 1787

This theorem is referenced by:  aaanv  42006  pm11.71  42015