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	$$⊢Ⅎy\phi$$
Assertion	sb8v	$$⊢\forall x\phi \leftrightarrow \forall y\left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		sb8v.nf	$$⊢Ⅎy\phi$$
2		nfs1v	$$⊢Ⅎx\left[y/x\right]\phi$$
3		sbequ12	$$⊢x=y\to \left(\phi \leftrightarrow \left[y/x\right]\phi \right)$$
4	1 2 3	cbvalv1	$$⊢\forall x\phi \leftrightarrow \forall y\left[y/x\right]\phi$$