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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	⊢ ¬ 𝜑
	Assertion	spfalw	⊢ ( ∀ 𝑥 𝜑 → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		spfalw.1	⊢ ¬ 𝜑
2	1	hbth	⊢ ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 )
3	2	spnfw	⊢ ( ∀ 𝑥 𝜑 → 𝜑 )