# Metamath Proof Explorer

## 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 ${⊢}\exists !{x}\phantom{\rule{.4em}{0ex}}{\phi }↔\left(\exists {x}\phantom{\rule{.4em}{0ex}}{\phi }\wedge {\exists }^{*}{x}\phantom{\rule{.4em}{0ex}}{\phi }\right)$

### Detailed syntax breakdown

Step Hyp Ref Expression
0 vx ${setvar}{x}$
1 wph ${wff}{\phi }$
2 1 0 weu ${wff}\exists !{x}\phantom{\rule{.4em}{0ex}}{\phi }$
3 1 0 wex ${wff}\exists {x}\phantom{\rule{.4em}{0ex}}{\phi }$
4 1 0 wmo ${wff}{\exists }^{*}{x}\phantom{\rule{.4em}{0ex}}{\phi }$
5 3 4 wa ${wff}\left(\exists {x}\phantom{\rule{.4em}{0ex}}{\phi }\wedge {\exists }^{*}{x}\phantom{\rule{.4em}{0ex}}{\phi }\right)$
6 2 5 wb ${wff}\left(\exists !{x}\phantom{\rule{.4em}{0ex}}{\phi }↔\left(\exists {x}\phantom{\rule{.4em}{0ex}}{\phi }\wedge {\exists }^{*}{x}\phantom{\rule{.4em}{0ex}}{\phi }\right)\right)$