Theorem sbco2v

Description: A composition law for substitution. Version of sbco2 with disjoint variable conditions, not requiring ax-13 , but ax-11 . (Contributed by NM, 30-Jun-1994) (Revised by Wolf Lammen, 29-Apr-2023)

	Ref	Expression
Hypothesis	sbco2v.1	$$⊢Ⅎz\phi$$
Assertion	sbco2v	$$⊢\left[y/z\right]\left[z/x\right]\phi \leftrightarrow \left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		sbco2v.1	$$⊢Ⅎz\phi$$
2	1	nfsbv	$$⊢Ⅎz\left[y/x\right]\phi$$
3		sbequ	$$⊢z=y\to \left(\left[z/x\right]\phi \leftrightarrow \left[y/x\right]\phi \right)$$
4	2 3	sbiev	$$⊢\left[y/z\right]\left[z/x\right]\phi \leftrightarrow \left[y/x\right]\phi$$