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 1966	. . . . 5 wff 𝑥 = 𝑦
6	1, 5	wi 4	. . . 4 wff (𝜑 → 𝑥 = 𝑦)
7	6, 2	wal 1539	. . 3 wff ∀𝑥(𝜑 → 𝑥 = 𝑦)
8	7, 4	wex 1781	. 2 wff ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)
9	3, 8	wb 205	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  2609  mopick  2620  2mo2  2642  rmoeq1  3413  mo2icl  3706  rmoanim  3884  axrep6  5285  moabex  5452  dffun3  6546  dffun3OLD  6547  dffun6f  6550  grothprim  10811  mobidvALT  35540  wl-cbvmotv  36186  wl-moteq  36187  wl-moae  36189  wl-mo2df  36239  wl-mo2t  36244  wl-mo3t  36245  sn-axrep5v  40848  sn-axprlem3  40849  dffrege115  42500  mof0  47152