Metamath Proof Explorer


Theorem sb1

Description: One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition ( sb5 ) or a nonfreeness hypothesis ( sb5f ). Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker sb1v when possible. (Contributed by NM, 13-May-1993) Revise df-sb . (Revised by Wolf Lammen, 21-Feb-2024) (New usage is discouraged.)

Ref Expression
Assertion sb1 yxφxx=yφ

Proof

Step Hyp Ref Expression
1 spsbe yxφxφ
2 pm3.2 x=yφx=yφ
3 2 aleximi xx=yxφxx=yφ
4 1 3 syl5 xx=yyxφxx=yφ
5 sb3b ¬xx=yyxφxx=yφ
6 5 biimpd ¬xx=yyxφxx=yφ
7 4 6 pm2.61i yxφxx=yφ