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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 ⊢ y x y x φ ↔ y x φ

Proof

Step Hyp Ref Expression
1 sbcom3 ⊢ y x x x φ ↔ y x y x φ
2 sbid ⊢ x x φ ↔ φ
3 2 sbbii ⊢ y x x x φ ↔ y x φ
4 1 3 bitr3i ⊢ y x y x φ ↔ y x φ