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Metamath Proof Explorer

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 ⊢ y x x y φ ↔ y x φ

Proof

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