Theorem bj-hbext

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

	Ref	Expression
Assertion	bj-hbext	⊢ ( ∀ 𝑦 ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜓 ) → ( ∃ 𝑦 𝜑 → ∀ 𝑥 ∃ 𝑦 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( ∀ 𝑦 ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑦 ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜓 ) )
2		sp	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜓 ) → ( 𝜑 → ∀ 𝑥 𝜓 ) )
3	1 2	bj-hbexd	⊢ ( ∀ 𝑦 ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜓 ) → ( ∃ 𝑦 𝜑 → ∀ 𝑥 ∃ 𝑦 𝜓 ) )