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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 = x -> A. y x = x )

Proof

Step	Hyp	Ref	Expression
1		ax-c9	\|- ( -. A. y y = x -> ( -. A. y y = x -> ( x = x -> A. y x = x ) ) )
2		ax7	\|- ( y = x -> ( y = x -> x = x ) )
3	2	pm2.43i	\|- ( y = x -> x = x )
4	3	alimi	\|- ( A. y y = x -> A. y x = x )
5	4	a1d	\|- ( A. y y = x -> ( x = x -> A. y x = x ) )
6	1 5 5	pm2.61ii	\|- ( x = x -> A. y x = x )