Theorem aaanh 1565

Description: Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)

Hypotheses

Ref	Expression
aaanh.1	⊢ (𝜑 → ∀𝑦𝜑)
aaanh.2	⊢ (𝜓 → ∀𝑥𝜓)

Assertion

Ref	Expression
aaanh	⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))

Proof of Theorem aaanh

Step	Hyp	Ref	Expression
1		aaanh.1	. . . 4 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	19.28h 1541	. . 3 ⊢ (∀𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑦𝜓))
3	2	albii 1446	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 ∧ ∀𝑦𝜓))
4		aaanh.2	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
5	4	hbal 1453	. . 3 ⊢ (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓)
6	5	19.27h 1539	. 2 ⊢ (∀𝑥(𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
7	3, 6	bitri 183	1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329

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 1423  ax-7 1424  ax-gen 1425  ax-4 1487

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  mo23  2040  2eu4  2092