Metamath Proof Explorer


Theorem hbequid

Description: Bound-variable hypothesis builder for x = x . This theorem tells us that any variable, including x , is effectively not free in x = x , even though x is technically free according to the traditional definition of free variable. (The proof does not use ax-c10 .) (Contributed by NM, 13-Jan-2011) (Proof shortened by Wolf Lammen, 23-Mar-2014) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion hbequid x=xyx=x

Proof

Step Hyp Ref Expression
1 ax-c9 ¬yy=x¬yy=xx=xyx=x
2 ax7 y=xy=xx=x
3 2 pm2.43i y=xx=x
4 3 alimi yy=xyx=x
5 4 a1d yy=xx=xyx=x
6 1 5 5 pm2.61ii x=xyx=x