Definition df-mo 2533

Description: Define the at-most-one quantifier. The expression ∃*𝑥𝜑 is read "there exists at most one 𝑥 such that 𝜑". This is also called the "uniqueness quantifier" but that expression is also used for the unique existential quantifier df-eu 2562, therefore we avoid that ambiguous name.

Notation of [BellMachover] p. 460, whose definition we show as mo3 2557. For other possible definitions see moeu 2576 and mo4 2559. (Contributed by Wolf Lammen, 27-May-2019.) Make this the definition (which used to be moeu 2576, while this definition was then proved as dfmo 2589). (Revised by BJ, 30-Sep-2022.)

Assertion

Ref	Expression
df-mo	⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Detailed syntax breakdown of Definition df-mo

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		vx	. . 3 setvar 𝑥
3	1, 2	wmo 2531	. 2 wff ∃*𝑥𝜑
4		vy	. . . . . 6 setvar 𝑦
5	2, 4	weq 1962	. . . . 5 wff 𝑥 = 𝑦
6	1, 5	wi 4	. . . 4 wff (𝜑 → 𝑥 = 𝑦)
7	6, 2	wal 1538	. . 3 wff ∀𝑥(𝜑 → 𝑥 = 𝑦)
8	7, 4	wex 1779	. 2 wff ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)
9	3, 8	wb 206	1 wff (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))

This definition is referenced by:  nexmo  2534  moim  2537  moimi  2538  nfmo1  2550  nfmod2  2551  nfmodv  2552  mof  2556  mo3  2557  mo4  2559  eu3v  2563  cbvmovw  2595  cbvmow  2596  sbmo  2607  mopick  2618  2mo2  2640  rmoeq1  3376  mo2icl  3674  rmoanim  3846  axrep6  5227  axrep6OLD  5228  moabex  5402  dffun3  6494  dffun6f  6497  grothprim  10728  cbvmodavw  36234  mobidvALT  36841  wl-cbvmotv  37497  wl-moteq  37498  wl-moae  37500  wl-mo2df  37554  wl-mo2t  37559  wl-mo3t  37560  sn-axrep5v  42199  sn-axprlem3  42200  dffrege115  43961  mof0  48832