This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem hbe1w

Description: Weak version of hbe1 . See comments for ax10w . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017)

		Ref	Expression
	Hypothesis	hbn1w.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
	Assertion	hbe1w	$⊢ \exists x φ \to \forall x \exists x φ$

Step	Hyp	Ref	Expression
1		hbn1w.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
2		df-ex	$⊢ \exists x φ \leftrightarrow \neg \forall x \neg φ$
3	1	notbid	$⊢ x = y \to (\neg φ \leftrightarrow \neg ψ)$
4	3	hbn1w	$⊢ \neg \forall x \neg φ \to \forall x \neg \forall x \neg φ$
5	2 4	hbxfrbi	$⊢ \exists x φ \to \forall x \exists x φ$