Definition df-eu 1948

Description: Define existential uniqueness, i.e. "there exists exactly one x such that ph." Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 1970, eu2 1989, eu3 1991, and eu5 1992 (which in some cases we show with a hypothesis ph -> A. y ph in place of a distinct variable condition on y and ph). Double uniqueness is tricky: E! x E! y ph does not mean "exactly one x and one y " (see 2eu4 2038). (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
df-eu	|- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )

Distinct variable groups:   x, y   ph, y

Allowed substitution hint:   ph( x)

Detailed syntax breakdown of Definition df-eu

Step	Hyp	Ref	Expression
1		wph	. . 3 wff ph
2		vx	. . 3 setvar x
3	1, 2	weu 1945	. 2 wff E! x ph
4		vy	. . . . . 6 setvar y
5	2, 4	weq 1435	. . . . 5 wff x = y
6	1, 5	wb 103	. . . 4 wff ( ph <-> x = y )
7	6, 2	wal 1285	. . 3 wff A. x ( ph <-> x = y )
8	7, 4	wex 1424	. 2 wff E. y A. x ( ph <-> x = y )
9	3, 8	wb 103	1 wff ( E! x ph <-> E. y A. x ( ph <-> x = y ) )

This definition is referenced by:  euf  1950  eubidh  1951  eubid  1952  hbeu1  1955  nfeu1  1956  sb8eu  1958  nfeudv  1960  nfeuv  1963  sb8euh  1968  exists1  2041  reu6  2795  euabsn2  3496  euotd  4057  iotauni  4960  iota1  4962  iotanul  4963  euiotaex  4964  iota4  4966  fv3  5293  eufnfv  5488