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  3384  mo2icl  3682  rmoanim  3854  axrep6  5238  axrep6OLD  5239  moabex  5414  dffun3  6512  dffun6f  6515  grothprim  10763  cbvmodavw  36211  mobidvALT  36818  wl-cbvmotv  37474  wl-moteq  37475  wl-moae  37477  wl-mo2df  37531  wl-mo2t  37536  wl-mo3t  37537  sn-axrep5v  42177  sn-axprlem3  42178  dffrege115  43940  mof0  48799