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	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	hbe1w	$$⊢\exists x\phi \to \forall x\exists x\phi$$

Proof

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