Theorem spfalw

Description: Version of sp when ph is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017) (Proof shortened by Wolf Lammen, 25-Dec-2017)

	Ref	Expression
Hypothesis	spfalw.1	$$⊢\neg \phi$$
Assertion	spfalw	$$⊢\forall x\phi \to \phi$$

Proof

Step	Hyp	Ref	Expression
1		spfalw.1	$$⊢\neg \phi$$
2	1	hbth	$$⊢\neg \phi \to \forall x\neg \phi$$
3	2	spnfw	$$⊢\forall x\phi \to \phi$$