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Metamath Proof Explorer

Theorem aev

Description: A "distinctor elimination" lemma with no disjoint variable conditions on variables in the consequent. (Contributed by NM, 8-Nov-2006) Remove dependency on ax-11 . (Revised by Wolf Lammen, 7-Sep-2018) Remove dependency on ax-13 , inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019) Remove dependency on ax-12 . (Revised by Wolf Lammen, 19-Mar-2021)

		Ref	Expression
	Assertion	aev	\|- ( A. x x = y -> A. z t = u )

Proof

Step	Hyp	Ref	Expression
1		aevlem	\|- ( A. x x = y -> A. v v = w )
2		aeveq	\|- ( A. v v = w -> t = u )
3	2	alrimiv	\|- ( A. v v = w -> A. z t = u )
4	1 3	syl	\|- ( A. x x = y -> A. z t = u )