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Metamath Proof Explorer

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	$⊢ \neg φ$
	Assertion	spfalw	$⊢ \forall x φ \to φ$

Proof

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