Theorem spnfw

Description: Weak version of sp . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017) (Proof shortened by Wolf Lammen, 13-Aug-2017)

	Ref	Expression
Hypothesis	spnfw.1	⊢ ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 )
Assertion	spnfw	⊢ ( ∀ 𝑥 𝜑 → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		spnfw.1	⊢ ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 )
2		idd	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜑 ) )
3	1 2	spimw	⊢ ( ∀ 𝑥 𝜑 → 𝜑 )