Theorem sbidm

Description: An idempotent law for substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 30-Jun-1994) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 13-Jul-2019) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		sbcom3	$$⊢\left[y/x\right]\left[x/x\right]\phi \leftrightarrow \left[y/x\right]\left[y/x\right]\phi$$
2		sbid	$$⊢\left[x/x\right]\phi \leftrightarrow \phi$$
3	2	sbbii	$$⊢\left[y/x\right]\left[x/x\right]\phi \leftrightarrow \left[y/x\right]\phi$$
4	1 3	bitr3i	$$⊢\left[y/x\right]\left[y/x\right]\phi \leftrightarrow \left[y/x\right]\phi$$