Theorem sbalexi

Description: Inference form of sbalex , avoiding ax-10 by using ax-gen . (Contributed by SN, 12-Aug-2025)

	Ref	Expression
Hypothesis	sbalexi.1	⊢ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 )
Assertion	sbalexi	⊢ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbalexi.1	⊢ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 )
2		ax12ev2	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) → ( 𝑥 = 𝑦 → 𝜑 ) )
3	1 2	ax-mp	⊢ ( 𝑥 = 𝑦 → 𝜑 )
4	3	ax-gen	⊢ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 )