P. 21 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2001-2100   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-mo 2001	Define "there exists at most one 𝑥 such that 𝜑." Here we define it in terms of existential uniqueness. Notation of [BellMachover] p. 460, whose definition we show as mo3 2051. For another possible definition see mo4 2058. (Contributed by NM, 5-Aug-1993.)
⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
Theorem	euf 2002*	A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)
⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
Theorem	eubidh 2003	Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
Theorem	eubid 2004	Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
Theorem	eubidv 2005*	Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
Theorem	eubii 2006	Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)
Theorem	hbeu1 2007	Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.)
⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
Theorem	nfeu1 2008	Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥∃!𝑥𝜑
Theorem	nfmo1 2009	Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥∃*𝑥𝜑
Theorem	sb8eu 2010	Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
Theorem	sb8mo 2011	Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)
Theorem	nfeudv 2012*	Deduction version of nfeu 2016. Similar to nfeud 2013 but has the additional constraint that 𝑥 and 𝑦 must be distinct. (Contributed by Jim Kingdon, 25-May-2018.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
Theorem	nfeud 2013	Deduction version of nfeu 2016. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 25-May-2018.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
Theorem	nfmod 2014	Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
Theorem	nfeuv 2015*	Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 2016 but has the additional constraint that 𝑥 and 𝑦 must be distinct. (Contributed by Jim Kingdon, 23-May-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃!𝑦𝜑
Theorem	nfeu 2016	Bound-variable hypothesis builder for existential uniqueness. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 23-May-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃!𝑦𝜑
Theorem	nfmo 2017	Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃*𝑦𝜑
Theorem	hbeu 2018	Bound-variable hypothesis builder for uniqueness. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Proof rewritten by Jim Kingdon, 24-May-2018.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∃!𝑦𝜑 → ∀𝑥∃!𝑦𝜑)
Theorem	hbeud 2019	Deduction version of hbeu 2018. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 25-May-2018.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → (∃!𝑦𝜓 → ∀𝑥∃!𝑦𝜓))
Theorem	sb8euh 2020	Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.)
⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
Theorem	cbveu 2021	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
Theorem	eu1 2022*	An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.)
⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
Theorem	euor 2023	Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 21-Oct-2005.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓))
Theorem	euorv 2024*	Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 23-Mar-1995.)
⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓))
Theorem	mo2n 2025*	There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (¬ ∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
Theorem	mon 2026	There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.)
⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
Theorem	euex 2027	Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)
Theorem	eumo0 2028*	Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)
⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ (∃!𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
Theorem	eumo 2029	Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)
⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)
Theorem	eumoi 2030	"At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
⊢ ∃!𝑥𝜑    ⇒   ⊢ ∃*𝑥𝜑
Theorem	mobidh 2031	Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
Theorem	mobid 2032	Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
Theorem	mobidv 2033*	Formula-building rule for "at most one" quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
Theorem	mobii 2034	Formula-building rule for "at most one" quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)
⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)
Theorem	hbmo1 2035	Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.)
⊢ (∃*𝑥𝜑 → ∀𝑥∃*𝑥𝜑)
Theorem	hbmo 2036	Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∃*𝑦𝜑 → ∀𝑥∃*𝑦𝜑)
Theorem	cbvmo 2037	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
Theorem	mo23 2038*	An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Theorem	mor 2039*	Converse of mo23 2038 with an additional ∃𝑥𝜑 condition. (Contributed by Jim Kingdon, 25-Jun-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃𝑥𝜑 → (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
Theorem	modc 2040*	Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (DECID ∃𝑥𝜑 → (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
Theorem	eu2 2041*	An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
Theorem	eu3h 2042*	An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
Theorem	eu3 2043*	An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
Theorem	eu5 2044	Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)
⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
Theorem	exmoeu2 2045	Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)
⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))
Theorem	moabs 2046	Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)
⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
Theorem	exmodc 2047	If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.)
⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ∨ ∃*𝑥𝜑))
Theorem	exmonim 2048	There is at most one of something which does not exist. Unlike exmodc 2047 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.)
⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
Theorem	mo2r 2049*	A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑)
Theorem	mo3h 2050*	Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that 𝑦 not occur in 𝜑 in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Theorem	mo3 2051*	Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that 𝑦 not occur in 𝜑 in place of our hypothesis. (Contributed by NM, 8-Mar-1995.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Theorem	mo2dc 2052*	Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (DECID ∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
Theorem	euan 2053	Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Theorem	euanv 2054*	Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.)
⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Theorem	euor2 2055	Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓))
Theorem	sbmo 2056*	Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)
Theorem	mo4f 2057*	"At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
Theorem	mo4 2058*	"At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
Theorem	eu4 2059*	Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
Theorem	exmoeudc 2060	Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.)
⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑)))
Theorem	moim 2061	"At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
Theorem	moimi 2062	"At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃*𝑥𝜓 → ∃*𝑥𝜑)
Theorem	moimv 2063*	Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)
⊢ (∃*𝑥(𝜑 → 𝜓) → (𝜑 → ∃*𝑥𝜓))
Theorem	euimmo 2064	Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))
Theorem	euim 2065	Add existential unique existential quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑))
Theorem	moan 2066	"At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)
⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜓 ∧ 𝜑))
Theorem	moani 2067	"At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)
⊢ ∃*𝑥𝜑    ⇒   ⊢ ∃*𝑥(𝜓 ∧ 𝜑)
Theorem	moor 2068	"At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)
⊢ (∃*𝑥(𝜑 ∨ 𝜓) → ∃*𝑥𝜑)
Theorem	mooran1 2069	"At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ((∃*𝑥𝜑 ∨ ∃*𝑥𝜓) → ∃*𝑥(𝜑 ∧ 𝜓))
Theorem	mooran2 2070	"At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ (∃*𝑥(𝜑 ∨ 𝜓) → (∃*𝑥𝜑 ∧ ∃*𝑥𝜓))
Theorem	moanim 2071	Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 3-Dec-2001.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
Theorem	moanimv 2072*	Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 23-Mar-1995.)
⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
Theorem	moaneu 2073	Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.)
⊢ ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)
Theorem	moanmo 2074	Nested at-most-one quantifiers. (Contributed by NM, 25-Jan-2006.)
⊢ ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)
Theorem	mopick 2075	"At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.)
⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
Theorem	eupick 2076	Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing 𝑥 such that 𝜑 is true, and there is also an 𝑥 (actually the same one) such that 𝜑 and 𝜓 are both true, then 𝜑 implies 𝜓 regardless of 𝑥. This theorem can be useful for eliminating existential quantifiers in a hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by NM, 10-Jul-1994.)
⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
Theorem	eupicka 2077	Version of eupick 2076 with closed formulas. (Contributed by NM, 6-Sep-2008.)
⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓))
Theorem	eupickb 2078	Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)
⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 ↔ 𝜓))
Theorem	eupickbi 2079	Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))
Theorem	mopick2 2080	"At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1610. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	moexexdc 2081	"At most one" double quantification. (Contributed by Jim Kingdon, 5-Jul-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (DECID ∃𝑥𝜑 → ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓)))
Theorem	euexex 2082	Existential uniqueness and "at most one" double quantification. (Contributed by Jim Kingdon, 28-Dec-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ ((∃!𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))
Theorem	2moex 2083	Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)
⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥𝜑)
Theorem	2euex 2084	Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑)
Theorem	2eumo 2085	Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.)
⊢ (∃!𝑥∃*𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
Theorem	2eu2ex 2086	Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
⊢ (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃𝑦𝜑)
Theorem	2moswapdc 2087	A condition allowing swap of "at most one" and existential quantifiers. (Contributed by Jim Kingdon, 6-Jul-2018.)
⊢ (DECID ∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)))
Theorem	2euswapdc 2088	A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Jim Kingdon, 7-Jul-2018.)
⊢ (DECID ∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑)))
Theorem	2exeu 2089	Double existential uniqueness implies double unique existential quantification. (Contributed by NM, 3-Dec-2001.)
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
Theorem	2eu4 2090*	This theorem provides us with a definition of double existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"). Naively one might think (incorrectly) that it could be defined by ∃!𝑥∃!𝑦𝜑. See 2exeu 2089 for a one-way implication. (Contributed by NM, 3-Dec-2001.)
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
Theorem	2eu7 2091	Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))
Theorem	euequ1 2092*	Equality has existential uniqueness. (Contributed by Stefan Allan, 4-Dec-2008.)
⊢ ∃!𝑥 𝑥 = 𝑦
Theorem	exists1 2093*	Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory. (Contributed by NM, 5-Apr-2004.)
⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
Theorem	exists2 2094	A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)
1.4.7  Aristotelian logic: Assertic syllogisms Model the Aristotelian assertic syllogisms using modern notation. This section shows that the Aristotelian assertic syllogisms can be proven with our axioms of logic, and also provides generally useful theorems. In antiquity Aristotelian logic and Stoic logic (see mptnan 1401) were the leading logical systems. Aristotelian logic became the leading system in medieval Europe; this section models this system (including later refinements to it). Aristotle defined syllogisms very generally ("a discourse in which certain (specific) things having been supposed, something different from the things supposed results of necessity because these things are so") Aristotle, Prior Analytics 24b18-20. However, in Prior Analytics he limits himself to categorical syllogisms that consist of three categorical propositions with specific structures. The syllogisms are the valid subset of the possible combinations of these structures. The medieval schools used vowels to identify the types of terms (a=all, e=none, i=some, and o=some are not), and named the different syllogisms with Latin words that had the vowels in the intended order. "There is a surprising amount of scholarly debate about how best to formalize Aristotle's syllogisms..." according to Aristotle's Modal Proofs: Prior Analytics A8-22 in Predicate Logic, Adriane Rini, Springer, 2011, ISBN 978-94-007-0049-9, page 28. For example, Lukasiewicz believes it is important to note that "Aristotle does not introduce singular terms or premisses into his system". Lukasiewicz also believes that Aristotelian syllogisms are predicates (having a true/false value), not inference rules: "The characteristic sign of an inference is the word 'therefore'... no syllogism is formulated by Aristotle primarily as an inference, but they are all implications." Jan Lukasiewicz, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, Second edition, Oxford, 1957, page 1-2. Lukasiewicz devised a specialized prefix notation for representing Aristotelian syllogisms instead of using standard predicate logic notation. We instead translate each Aristotelian syllogism into an inference rule, and each rule is defined using standard predicate logic notation and predicates. The predicates are represented by wff variables that may depend on the quantified variable 𝑥. Our translation is essentially identical to the one use in Rini page 18, Table 2 "Non-Modal Syllogisms in Lower Predicate Calculus (LPC)", which uses standard predicate logic with predicates. Rini states, "the crucial point is that we capture the meaning Aristotle intends, and the method by which we represent that meaning is less important." There are two differences: we make the existence criteria explicit, and we use 𝜑, 𝜓, and 𝜒 in the order they appear (a common Metamath convention). Patzig also uses standard predicate logic notation and predicates (though he interprets them as conditional propositions, not as inference rules); see Gunther Patzig, Aristotle's Theory of the Syllogism second edition, 1963, English translation by Jonathan Barnes, 1968, page 38. Terms such as "all" and "some" are translated into predicate logic using the aproach devised by Frege and Russell. "Frege (and Russell) devised an ingenious procedure for regimenting binary quantifiers like "every" and "some" in terms of unary quantifiers like "everything" and "something": they formalized sentences of the form "Some A is B" and "Every A is B" as exists x (Ax and Bx) and all x (Ax implies Bx), respectively." "Quantifiers and Quantification", Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/quantification/ 1401. See Principia Mathematica page 22 and *10 for more information (especially *10.3 and *10.26). Expressions of the form "no 𝜑 is 𝜓 " are consistently translated as ∀𝑥(𝜑 → ¬ 𝜓). These can also be expressed as ¬ ∃𝑥(𝜑 ∧ 𝜓), per alinexa 1582. We translate "all 𝜑 is 𝜓 " to ∀𝑥(𝜑 → 𝜓), "some 𝜑 is 𝜓 " to ∃𝑥(𝜑 ∧ 𝜓), and "some 𝜑 is not 𝜓 " to ∃𝑥(𝜑 ∧ ¬ 𝜓). It is traditional to use the singular verb "is", not the plural verb "are", in the generic expressions. By convention the major premise is listed first. In traditional Aristotelian syllogisms the predicates have a restricted form ("x is a ..."); those predicates could be modeled in modern notation by constructs such as 𝑥 = 𝐴, 𝑥 ∈ 𝐴, or 𝑥 ⊆ 𝐴. Here we use wff variables instead of specialized restricted forms. This generalization makes the syllogisms more useful in more circumstances. In addition, these expressions make it clearer that the syllogisms of Aristolean logic are the forerunners of predicate calculus. If we used restricted forms like 𝑥 ∈ 𝐴 instead, we would not only unnecessarily limit their use, but we would also need to use set and class axioms, making their relationship to predicate calculus less clear. There are some widespread misconceptions about the existential assumptions made by Aristotle (aka "existential import"). Aristotle was not trying to develop something exactly corresponding to modern logic. Aristotle devised "a companion-logic for science. He relegates fictions like fairy godmothers and mermaids and unicorns to the realms of poetry and literature. In his mind, they exist outside the ambit of science. This is why he leaves no room for such nonexistent entities in his logic. This is a thoughtful choice, not an inadvertent omission. Technically, Aristotelian science is a search for definitions, where a definition is "a phrase signifying a thing's essence." (Topics, I.5.102a37, Pickard-Cambridge.)... Because nonexistent entities cannot be anything, they do not, in Aristotle's mind, possess an essence... This is why he leaves no place for fictional entities like goat-stags (or unicorns)." Source: Louis F. Groarke, "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy (A Peer-Reviewed Academic Resource), http://www.iep.utm.edu/aris-log/ 1582. Thus, some syllogisms have "extra" existence hypotheses that do not directly appear in Aristotle's original materials (since they were always assumed); they are added where they are needed. This affects barbari 2099, celaront 2100, cesaro 2105, camestros 2106, felapton 2111, darapti 2112, calemos 2116, fesapo 2117, and bamalip 2118. These are only the assertic syllogisms. Aristotle also defined modal syllogisms that deal with modal qualifiers such as "necessarily" and "possibly". Historically Aristotelian modal syllogisms were not as widely used. For more about modal syllogisms in a modern context, see Rini as well as Aristotle's Modal Syllogistic by Marko Malink, Harvard University Press, November 2013. We do not treat them further here. Aristotelean logic is essentially the forerunner of predicate calculus (as well as set theory since it discusses membership in groups), while Stoic logic is essentially the forerunner of propositional calculus.
Theorem	barbara 2095	"Barbara", one of the fundamental syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and all 𝜒 is 𝜑, therefore all 𝜒 is 𝜓. (In Aristotelian notation, AAA-1: MaP and SaM therefore SaP.) For example, given "All men are mortal" and "Socrates is a man", we can prove "Socrates is mortal". If H is the set of men, M is the set of mortal beings, and S is Socrates, these word phrases can be represented as ∀𝑥(𝑥 ∈ 𝐻 → 𝑥 ∈ 𝑀) (all men are mortal) and ∀𝑥(𝑥 = 𝑆 → 𝑥 ∈ 𝐻) (Socrates is a man) therefore ∀𝑥(𝑥 = 𝑆 → 𝑥 ∈ 𝑀) (Socrates is mortal). Russell and Whitehead note that the "syllogism in Barbara is derived..." from syl 14. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1614. There are a legion of sources for Barbara, including http://www.friesian.com/aristotl.htm 1614, http://plato.stanford.edu/entries/aristotle-logic/ 1614, and https://en.wikipedia.org/wiki/Syllogism 1614. (Contributed by David A. Wheeler, 24-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    ⇒   ⊢ ∀𝑥(𝜒 → 𝜓)
Theorem	celarent 2096	"Celarent", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and all 𝜒 is 𝜑, therefore no 𝜒 is 𝜓. (In Aristotelian notation, EAE-1: MeP and SaM therefore SeP.) For example, given the "No reptiles have fur" and "All snakes are reptiles", therefore "No snakes have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    ⇒   ⊢ ∀𝑥(𝜒 → ¬ 𝜓)
Theorem	darii 2097	"Darii", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜒 is 𝜑, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AII-1: MaP and SiM therefore SiP.) For example, given "All rabbits have fur" and "Some pets are rabbits", therefore "Some pets have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ 𝜑)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	ferio 2098	"Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜑, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EIO-1: MeP and SiM therefore SoP.) For example, given "No homework is fun" and "Some reading is homework", therefore "Some reading is not fun". This is essentially a logical axiom in Aristotelian logic. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ 𝜑)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	barbari 2099	"Barbari", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AAI-1: MaP and SaM therefore SiP.) For example, given "All men are mortal", "All Greeks are men", and "Greeks exist", therefore "Some Greeks are mortal". Note the existence hypothesis (to prove the "some" in the conclusion). Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 30-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	celaront 2100	"Celaront", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EAO-1: MeP and SaM therefore SoP.) For example, given "No reptiles have fur", "All snakes are reptiles.", and "Snakes exist.", prove "Some snakes have no fur". Note the existence hypothesis. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)