Definition df-eu 2089

Description: Define existential uniqueness, i.e., "there exists exactly one 𝑥 such that 𝜑". Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 2111, eu2 2131, eu3 2133, and eu5 2134 (which in some cases we show with a hypothesis 𝜑 → ∀𝑦𝜑 in place of a distinct variable condition on 𝑦 and 𝜑). Double uniqueness is tricky: ∃!𝑥∃!𝑦𝜑 does not mean "exactly one 𝑥 and one 𝑦 " (see 2eu4 2180). (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
df-eu	⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Detailed syntax breakdown of Definition df-eu

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		vx	. . 3 setvar 𝑥
3	1, 2	weu 2086	. 2 wff ∃!𝑥𝜑
4		vy	. . . . . 6 setvar 𝑦
5	2, 4	weq 1556	. . . . 5 wff 𝑥 = 𝑦
6	1, 5	wb 105	. . . 4 wff (𝜑 ↔ 𝑥 = 𝑦)
7	6, 2	wal 1400	. . 3 wff ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
8	7, 4	wex 1545	. 2 wff ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
9	3, 8	wb 105	1 wff (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))

This definition is used by:  euf  2091  eubidh  2092  eubid  2093  hbeu1  2096  nfeu1  2097  sb8eu  2099  nfeudv  2101  nfeuv  2104  sb8euh  2109  exists1  2183  cbvreuvw  2792  reu6  3015  euabsn2  3780  euotd  4395  iotauni  5350  iota1  5352  iotanul  5353  euiotaex  5354  iota4  5357  eliotaeu  5366  fv3  5718  eufnfv  5949