Theorem aaan 2323

Description: Distribute universal quantifiers. (Contributed by NM, 12-Aug-1993.) Avoid ax-10 2130. (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 1867	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓))
2		aaan.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
3	2	19.3 2191	. . . 4 ⊢ (∀𝑦𝜑 ↔ 𝜑)
4	3	albii 1814	. . 3 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑥𝜑)
5		alcom 2149	. . . 4 ⊢ (∀𝑥∀𝑦𝜓 ↔ ∀𝑦∀𝑥𝜓)
6		aaan.2	. . . . . 6 ⊢ Ⅎ𝑥𝜓
7	6	19.3 2191	. . . . 5 ⊢ (∀𝑥𝜓 ↔ 𝜓)
8	7	albii 1814	. . . 4 ⊢ (∀𝑦∀𝑥𝜓 ↔ ∀𝑦𝜓)
9	5, 8	bitri 275	. . 3 ⊢ (∀𝑥∀𝑦𝜓 ↔ ∀𝑦𝜓)
10	4, 9	anbi12i 627	. 2 ⊢ ((∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
11	1, 10	bitri 275	1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))

Syntax hints:   ↔ wb 205   ∧ wa 395  ∀wal 1532  Ⅎwnf 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-11 2147  ax-12 2167

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-nf 1779

This theorem is referenced by:  aaanv  43816  pm11.71  43825