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Metamath Proof Explorer

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	⊢ ( ∀ 𝑥 𝜑 → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ax-5	⊢ ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 )
2	1	spnfw	⊢ ( ∀ 𝑥 𝜑 → 𝜑 )