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Metamath Proof Explorer

Theorem moeu

Description: Uniqueness is equivalent to existence implying unique existence. Alternate definition of the at-most-one quantifier, in terms of the existential quantifier and the unique existential quantifier. (Contributed by NM, 8-Mar-1995) This used to be the definition of the at-most-one quantifier, while df-mo was then proved as dfmo . (Revised by BJ, 30-Sep-2022)

		Ref	Expression
	Assertion	moeu	\|- ( E* x ph <-> ( E. x ph -> E! x ph ) )

Proof

Step	Hyp	Ref	Expression
1		moabs	\|- ( E* x ph <-> ( E. x ph -> E* x ph ) )
2		exmoeub	\|- ( E. x ph -> ( E* x ph <-> E! x ph ) )
3	2	pm5.74i	\|- ( ( E. x ph -> E* x ph ) <-> ( E. x ph -> E! x ph ) )
4	1 3	bitri	\|- ( E* x ph <-> ( E. x ph -> E! x ph ) )