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	$$⊢x=x\to \forall yx=x$$

Proof

Step	Hyp	Ref	Expression
1		ax-c9	$$⊢\neg \forall yy=x\to \left(\neg \forall yy=x\to \left(x=x\to \forall yx=x\right)\right)$$
2		ax7	$$⊢y=x\to \left(y=x\to x=x\right)$$
3	2	pm2.43i	$$⊢y=x\to x=x$$
4	3	alimi	$$⊢\forall yy=x\to \forall yx=x$$
5	4	a1d	$$⊢\forall yy=x\to \left(x=x\to \forall yx=x\right)$$
6	1 5 5	pm2.61ii	$$⊢x=x\to \forall yx=x$$