Theorem sbco

Description: A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 . See sbcov for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 14-May-1993) (Proof shortened by Wolf Lammen, 21-Sep-2018) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		sbcom3	$$⊢\left[y/x\right]\left[x/y\right]\phi \leftrightarrow \left[y/x\right]\left[y/y\right]\phi$$
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	$$⊢\left[y/x\right]\left[x/y\right]\phi \leftrightarrow \left[y/x\right]\phi$$