Theorem hbn1w

Description: Weak version of hbn1 . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017)

	Ref	Expression
Hypothesis	hbn1w.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	hbn1w	$$⊢\neg \forall x\phi \to \forall x\neg \forall x\phi$$

Proof

Step	Hyp	Ref	Expression
1		hbn1w.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		ax-5	$$⊢\forall x\phi \to \forall y\forall x\phi$$
3		ax-5	$$⊢\neg \psi \to \forall x\neg \psi$$
4		ax-5	$$⊢\forall y\psi \to \forall x\forall y\psi$$
5		ax-5	$$⊢\neg \phi \to \forall y\neg \phi$$
6		ax-5	$$⊢\neg \forall y\psi \to \forall x\neg \forall y\psi$$
7	2 3 4 5 6 1	hbn1fw	$$⊢\neg \forall x\phi \to \forall x\neg \forall x\phi$$