Metamath Proof Explorer


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 yxyxφyxφ

Proof

Step Hyp Ref Expression
1 sbcom3 yxxxφyxyxφ
2 sbid xxφφ
3 2 sbbii yxxxφyxφ
4 1 3 bitr3i yxyxφyxφ