Theorem bj-nfext

Description: Closed form of nfex . (Contributed by BJ, 10-Oct-2019)

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

Proof

Step	Hyp	Ref	Expression
1		nf5	⊢ ( Ⅎ 𝑦 𝜑 ↔ ∀ 𝑦 ( 𝜑 → ∀ 𝑦 𝜑 ) )
2	1	biimpi	⊢ ( Ⅎ 𝑦 𝜑 → ∀ 𝑦 ( 𝜑 → ∀ 𝑦 𝜑 ) )
3	2	alimi	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ∀ 𝑦 𝜑 ) )
4		nfa2	⊢ Ⅎ 𝑦 ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ∀ 𝑦 𝜑 )
5		bj-hbext	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑦 ∃ 𝑥 𝜑 ) )
6	4 5	alrimi	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ∀ 𝑦 𝜑 ) → ∀ 𝑦 ( ∃ 𝑥 𝜑 → ∀ 𝑦 ∃ 𝑥 𝜑 ) )
7	3 6	syl	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → ∀ 𝑦 ( ∃ 𝑥 𝜑 → ∀ 𝑦 ∃ 𝑥 𝜑 ) )
8		nf5	⊢ ( Ⅎ 𝑦 ∃ 𝑥 𝜑 ↔ ∀ 𝑦 ( ∃ 𝑥 𝜑 → ∀ 𝑦 ∃ 𝑥 𝜑 ) )
9	7 8	sylibr	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → Ⅎ 𝑦 ∃ 𝑥 𝜑 )