Theorem hbn1fw

Description: Weak version of ax-10 from which we can prove any ax-10 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017) (Proof shortened by Wolf Lammen, 28-Feb-2018)

	Ref	Expression
Hypotheses	hbn1fw.1	$$⊢\forall x\phi \to \forall y\forall x\phi$$
	hbn1fw.2	$$⊢\neg \psi \to \forall x\neg \psi$$
	hbn1fw.3	$$⊢\forall y\psi \to \forall x\forall y\psi$$
	hbn1fw.4	$$⊢\neg \phi \to \forall y\neg \phi$$
	hbn1fw.5	$$⊢\neg \forall y\psi \to \forall x\neg \forall y\psi$$
	hbn1fw.6	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	hbn1fw	$$⊢\neg \forall x\phi \to \forall x\neg \forall x\phi$$

Proof

Step	Hyp	Ref	Expression
1		hbn1fw.1	$$⊢\forall x\phi \to \forall y\forall x\phi$$
2		hbn1fw.2	$$⊢\neg \psi \to \forall x\neg \psi$$
3		hbn1fw.3	$$⊢\forall y\psi \to \forall x\forall y\psi$$
4		hbn1fw.4	$$⊢\neg \phi \to \forall y\neg \phi$$
5		hbn1fw.5	$$⊢\neg \forall y\psi \to \forall x\neg \forall y\psi$$
6		hbn1fw.6	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
7	1 2 3 4 6	cbvalw	$$⊢\forall x\phi \leftrightarrow \forall y\psi$$
8	7	notbii	$$⊢\neg \forall x\phi \leftrightarrow \neg \forall y\psi$$
9	8 5	hbxfrbi	$$⊢\neg \forall x\phi \to \forall x\neg \forall x\phi$$