Theorem sbco2vv

Description: A composition law for substitution. Version of sbco2 with disjoint variable conditions and fewer axioms. (Contributed by NM, 30-Jun-1994) (Revised by BJ, 22-Dec-2020) (Proof shortened by Wolf Lammen, 29-Apr-2023)

	Ref	Expression
Assertion	sbco2vv	$$⊢\left[y/z\right]\left[z/x\right]\phi \leftrightarrow \left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		sbequ	$$⊢z=w\to \left(\left[z/x\right]\phi \leftrightarrow \left[w/x\right]\phi \right)$$
2		sbequ	$$⊢w=y\to \left(\left[w/x\right]\phi \leftrightarrow \left[y/x\right]\phi \right)$$
3	1 2	sbievw2	$$⊢\left[y/z\right]\left[z/x\right]\phi \leftrightarrow \left[y/x\right]\phi$$