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	|- ( [ y / x ] [ y / x ] ph <-> [ y / x ] ph )

Proof

Step	Hyp	Ref	Expression
1		sbcom3	|- ( [ y / x ] [ x / x ] ph <-> [ y / x ] [ y / x ] ph )
2		sbid	|- ( [ x / x ] ph <-> ph )
3	2	sbbii	|- ( [ y / x ] [ x / x ] ph <-> [ y / x ] ph )
4	1 3	bitr3i	|- ( [ y / x ] [ y / x ] ph <-> [ y / x ] ph )