Theorem hbe1a

Description: Dual statement of hbe1 . Modified version of axc7e with a universally quantified consequent. (Contributed by Wolf Lammen, 15-Sep-2021)

	Ref	Expression
Assertion	hbe1a	⊢ ( ∃ 𝑥 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		df-ex	⊢ ( ∃ 𝑥 ∀ 𝑥 𝜑 ↔ ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 )
2		hbn1	⊢ ( ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 𝜑 )
3	2	con1i	⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 )
4	1 3	sylbi	⊢ ( ∃ 𝑥 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 )