Theorem bj-nfalt

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

	Ref	Expression
Assertion	bj-nfalt	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → Ⅎ 𝑦 ∀ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		bj-hbalt	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) )
2	1	alimi	⊢ ( ∀ 𝑦 ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ∀ 𝑦 ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) )
3	2	alcoms	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ∀ 𝑦 𝜑 ) → ∀ 𝑦 ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) )
4		nf5	⊢ ( Ⅎ 𝑦 𝜑 ↔ ∀ 𝑦 ( 𝜑 → ∀ 𝑦 𝜑 ) )
5	4	albii	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ∀ 𝑦 𝜑 ) )
6		nf5	⊢ ( Ⅎ 𝑦 ∀ 𝑥 𝜑 ↔ ∀ 𝑦 ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) )
7	3 5 6	3imtr4i	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → Ⅎ 𝑦 ∀ 𝑥 𝜑 )