Theorem bj-nfalt 37188

Description: Closed form of nfal 2355. (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 37155	. . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
2	1	alimi 1831	. . 3 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
3	2	alcoms 2192	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
4		nf5 2316	. . 3 ⊢ (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑))
5	4	albii 1839	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 ↔ ∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑))
6		nf5 2316	. 2 ⊢ (Ⅎ𝑦∀𝑥𝜑 ↔ ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
7	3, 5, 6	3imtr4i 294	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1558  Ⅎwnf 1803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-10 2175  ax-11 2191  ax-12 2212

This theorem depends on definitions:  df-bi 209  df-or 859  df-ex 1800  df-nf 1804

This theorem is referenced by:  bj-dvelimdv1  37337