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	$$⊢\exists x\forall x\phi \to \forall x\phi$$

Proof

Step	Hyp	Ref	Expression
1		df-ex	$$⊢\exists x\forall x\phi \leftrightarrow \neg \forall x\neg \forall x\phi$$
2		hbn1	$$⊢\neg \forall x\phi \to \forall x\neg \forall x\phi$$
3	2	con1i	$$⊢\neg \forall x\neg \forall x\phi \to \forall x\phi$$
4	1 3	sylbi	$$⊢\exists x\forall x\phi \to \forall x\phi$$