Metamath Proof Explorer


Theorem equs5e

Description: A property related to substitution that unlike equs5 does not require a distinctor antecedent. This proof uses ax12 , see equs5eALT for an alternative one using ax-12 but not ax13 . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Feb-2007) (Proof shortened by Wolf Lammen, 15-Jan-2018) (New usage is discouraged.)

Ref Expression
Assertion equs5e xx=yφxx=yyφ

Proof

Step Hyp Ref Expression
1 nfa1 xxx=yyφ
2 ax12 x=yyyφxx=yyφ
3 hbe1 yφyyφ
4 3 19.23bi φyyφ
5 2 4 impel x=yφxx=yyφ
6 1 5 exlimi xx=yφxx=yyφ