Theorem aaanOLD 2322

Description: Obsolete version of aaan 2321 as of 21-Nov-2024. (Contributed by NM, 12-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
aaan.1	⊢ Ⅎ𝑦𝜑
aaan.2	⊢ Ⅎ𝑥𝜓

Assertion

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

Proof of Theorem aaanOLD

Step	Hyp	Ref	Expression
1		aaan.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2	1	19.28 2213	. . 3 ⊢ (∀𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑦𝜓))
3	2	albii 1813	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 ∧ ∀𝑦𝜓))
4		aaan.2	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	nfal 2310	. . 3 ⊢ Ⅎ𝑥∀𝑦𝜓
6	5	19.27 2212	. 2 ⊢ (∀𝑥(𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
7	3, 6	bitri 275	1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))

Syntax hints:   ↔ wb 205   ∧ wa 395  ∀wal 1531  Ⅎwnf 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778

This theorem is referenced by: (None)