Definition df-eu 2045

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 2067, eu2 2086, eu3 2088, and eu5 2089 (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 2135). (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 2042	. 2 wff ∃!𝑥𝜑
4		vy	. . . . . 6 setvar 𝑦
5	2, 4	weq 1514	. . . . 5 wff 𝑥 = 𝑦
6	1, 5	wb 105	. . . 4 wff (𝜑 ↔ 𝑥 = 𝑦)
7	6, 2	wal 1362	. . 3 wff ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
8	7, 4	wex 1503	. 2 wff ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
9	3, 8	wb 105	1 wff (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))

This definition is referenced by:  euf  2047  eubidh  2048  eubid  2049  hbeu1  2052  nfeu1  2053  sb8eu  2055  nfeudv  2057  nfeuv  2060  sb8euh  2065  exists1  2138  cbvreuvw  2732  reu6  2950  euabsn2  3688  euotd  4284  iotauni  5228  iota1  5230  iotanul  5231  euiotaex  5232  iota4  5235  eliotaeu  5244  fv3  5578  eufnfv  5790