Theorem nfalt 1566

Description: Closed form of nfal 1564. (Contributed by Jim Kingdon, 11-May-2018.)

Assertion

Ref	Expression
nfalt	⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥∀𝑦𝜑)

Proof of Theorem nfalt

Step	Hyp	Ref	Expression
1		alim 1445	. . . 4 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑))
2		alcom 1466	. . . 4 ⊢ (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)
3	1, 2	syl6ib 160	. . 3 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
4	3	alimi 1443	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
5		df-nf 1449	. . . 4 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
6	5	albii 1458	. . 3 ⊢ (∀𝑦Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑))
7		alcom 1466	. . 3 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
8	6, 7	bitri 183	. 2 ⊢ (∀𝑦Ⅎ𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑))
9		df-nf 1449	. 2 ⊢ (Ⅎ𝑥∀𝑦𝜑 ↔ ∀𝑥(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))
10	4, 8, 9	3imtr4i 200	1 ⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥∀𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1341  Ⅎwnf 1448

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 1435  ax-7 1436  ax-gen 1437

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  dvelimor  2006