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 yxφyxx=yφ

Proof

Step Hyp Ref Expression
1 equsb1 yxx=y
2 1 a1bi yxφyxx=yyxφ
3 sbim yxx=yφyxx=yyxφ
4 2 3 bitr4i yxφyxx=yφ