Theorem sbcov

Description: A composition law for substitution. Version of sbco with a disjoint variable condition using fewer axioms. (Contributed by NM, 14-May-1993) (Revised by Gino Giotto, 7-Aug-2023)

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

Proof

Step	Hyp	Ref	Expression
1		sbcom3vv	$$⊢\left[y/x\right]\left[x/y\right]\phi \leftrightarrow \left[y/x\right]\left[y/y\right]\phi$$
2		sbid	$$⊢\left[y/y\right]\phi \leftrightarrow \phi$$
3	2	sbbii	$$⊢\left[y/x\right]\left[y/y\right]\phi \leftrightarrow \left[y/x\right]\phi$$
4	1 3	bitri	$$⊢\left[y/x\right]\left[x/y\right]\phi \leftrightarrow \left[y/x\right]\phi$$