Theorem spvw

Description: Version of sp when x does not occur in ph . Converse of ax-5 . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017) (Proof shortened by Wolf Lammen, 4-Dec-2017) Shorten 19.3v . (Revised by Wolf Lammen, 20-Oct-2023)

	Ref	Expression
Assertion	spvw	$$⊢\forall x\phi \to \phi$$

Proof

Step	Hyp	Ref	Expression
1		ax-5	$$⊢\neg \phi \to \forall x\neg \phi$$
2	1	spnfw	$$⊢\forall x\phi \to \phi$$