Theorem spALT

Description: sp can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023) (Proof modification is discouraged.) Use sp instead. (New usage is discouraged.)

	Ref	Expression
Assertion	spALT	⊢ ( ∀ 𝑥 𝜑 → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ax-1	⊢ ( ∀ 𝑥 𝜑 → ( 𝑥 = 𝑦 → ∀ 𝑥 𝜑 ) )
2	1	axc4i	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → ∀ 𝑥 𝜑 ) )
3		axc10	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ∀ 𝑥 𝜑 ) → 𝜑 )
4	2 3	syl	⊢ ( ∀ 𝑥 𝜑 → 𝜑 )