Definition df-eu

Description: Define the existential uniqueness quantifier. This expresses unique existence, or existential uniqueness, which is the conjunction of existence ( df-ex ) and uniqueness ( df-mo ). The expression E! x ph is read "there exists exactly one x such that ph " or "there exists a unique x such that ph ". This is also called the "uniqueness quantifier" but that expression is also used for the at-most-one quantifier df-mo , therefore we avoid that ambiguous name.

Definition 10.1 of BellMachover p. 97; also Definition *14.02 of WhiteheadRussell p. 175. Other possible definitions are given by eu1 , eu2 , eu3v , and eu6 . As for double unique existence, beware that the expression E! x E! y ph means "there exists a unique x such that there exists a unique y such that ph " which is a weaker property than "there exists exactly one x and one y such that ph " (see 2eu4 ). (Contributed by NM, 12-Aug-1993) Make this the definition (which used to be eu6 , while this definition was then proved as dfeu ). (Revised by BJ, 30-Sep-2022)

	Ref	Expression
Assertion	df-eu	$$⊢\exists !x\phi \leftrightarrow \left(\exists x\phi \wedge {\exists }^{*}x\phi \right)$$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		vx	$$setvarx$$
1		wph	$$wff\phi$$
2	1 0	weu	$$wff\exists !x\phi$$
3	1 0	wex	$$wff\exists x\phi$$
4	1 0	wmo	$$wff{\exists }^{*}x\phi$$
5	3 4	wa	$$wff\left(\exists x\phi \wedge {\exists }^{*}x\phi \right)$$
6	2 5	wb	$$wff\left(\exists !x\phi \leftrightarrow \left(\exists x\phi \wedge {\exists }^{*}x\phi \right)\right)$$