Theorem bj-nfalt 31889

Description: Closed form of nfal 2139. (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 31858	. . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
2	1	alimi 1730	. . 3 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
3	2	alcoms 2022	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑) → ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
4		nf5 2102	. . 3 ⊢ (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑))
5	4	albii 1737	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 ↔ ∀𝑥∀𝑦(𝜑 → ∀𝑦𝜑))
6		nf5 2102	. 2 ⊢ (Ⅎ𝑦∀𝑥𝜑 ↔ ∀𝑦(∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
7	3, 5, 6	3imtr4i 280	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1473  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701

This theorem is referenced by:  bj-dvelimdv1  32028