Theorem bj-nfalt 34872

Description: Closed form of nfal 2320. (Contributed by BJ, 2-May-2019.)

Assertion

Ref	Expression
bj-nfalt	⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑)

Proof of Theorem bj-nfalt

Step	Hyp	Ref	Expression
1		bj-hbalt 34842	. . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
2	1	alimi 1817	. . 3 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
3	2	alcoms 2158	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
4		nf5 2282	. . 3 ⊢ (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑))
5	4	albii 1825	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 ↔ ∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑))
6		nf5 2282	. 2 ⊢ (Ⅎ𝑦∀𝑥𝜑 ↔ ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
7	3, 5, 6	3imtr4i 291	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1539  Ⅎwnf 1789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-10 2140  ax-11 2157  ax-12 2174

This theorem depends on definitions:  df-bi 206  df-or 844  df-ex 1786  df-nf 1790

This theorem is referenced by:  bj-dvelimdv1  35015