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Metamath Proof Explorer

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	$⊢ \exists x \forall x φ \to \forall x φ$

Proof

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