Metamath Proof Explorer


Theorem sbequ8

Description: Elimination of equality from antecedent after substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 5-Aug-1993) Reduce dependencies on axioms. (Revised by Wolf Lammen, 28-Jul-2018) Revise df-sb . (Revised by Wolf Lammen, 28-Jul-2023) (New usage is discouraged.)

Ref Expression
Assertion sbequ8 y x φ y x x = y φ

Proof

Step Hyp Ref Expression
1 equsb1 y x x = y
2 1 a1bi y x φ y x x = y y x φ
3 sbim y x x = y φ y x x = y y x φ
4 2 3 bitr4i y x φ y x x = y φ