Metamath Proof Explorer


Theorem equs5a

Description: A property related to substitution that unlike equs5 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 . This proof uses ax12 , see equs5aALT for an alternative one using ax-12 but not ax13 . Usage of the weaker equs5av is preferred, which uses ax12v2 , but not ax-13 . (Contributed by NM, 2-Feb-2007) (New usage is discouraged.)

Ref Expression
Assertion equs5a xx=yyφxx=yφ

Proof

Step Hyp Ref Expression
1 nfa1 xxx=yφ
2 ax12 x=yyφxx=yφ
3 2 imp x=yyφxx=yφ
4 1 3 exlimi xx=yyφxx=yφ