Definition df-eu 2041

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 2063, eu2 2082, eu3 2084, and eu5 2085 (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 2131). (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 2038	. 2 wff E! x ph
4		vy	. . . . . 6 setvar y
5	2, 4	weq 1514	. . . . 5 wff x = y
6	1, 5	wb 105	. . . 4 wff ( ph <-> x = y )
7	6, 2	wal 1362	. . 3 wff A. x ( ph <-> x = y )
8	7, 4	wex 1503	. 2 wff E. y A. x ( ph <-> x = y )
9	3, 8	wb 105	1 wff ( E! x ph <-> E. y A. x ( ph <-> x = y ) )

This definition is referenced by:  euf  2043  eubidh  2044  eubid  2045  hbeu1  2048  nfeu1  2049  sb8eu  2051  nfeudv  2053  nfeuv  2056  sb8euh  2061  exists1  2134  cbvreuvw  2724  reu6  2941  euabsn2  3676  euotd  4272  iotauni  5208  iota1  5210  iotanul  5211  euiotaex  5212  iota4  5215  eliotaeu  5224  fv3  5557  eufnfv  5767