Theorem sb8v

Description: Substitution of variable in universal quantifier. Version of sb8 with a disjoint variable condition, not requiring ax-13 . (Contributed by NM, 16-May-1993) (Revised by Wolf Lammen, 19-Jan-2023)

	Ref	Expression
Hypothesis	sb8v.nf	⊢ Ⅎ 𝑦 𝜑
Assertion	sb8v	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sb8v.nf	⊢ Ⅎ 𝑦 𝜑
2		nfs1v	⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑
3		sbequ12	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
4	1 2 3	cbvalv1	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )