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
Theorem	exsb 2001*	An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)
|- ( E. x ph <-> E. y A. x ( x = y -> ph ) )
Theorem	2exsb 2002*	An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.)
|- ( E. x E. y ph <-> E. z E. w A. x A. y ( ( x = z /\ y = w ) -> ph ) )
Theorem	dvelimALT 2003*	Version of dvelim 2010 that doesn't use ax-10 1498. Because it has different distinct variable constraints than dvelim 2010 and is used in important proofs, it would be better if it had a name which does not end in ALT (ideally more close to set.mm naming). (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ph -> A. x ph )   &    |- ( z = y -> ( ph <-> ps ) )   =>    |- ( -. A. x x = y -> ( ps -> A. x ps ) )
Theorem	dvelimfv 2004*	Like dvelimf 2008 but with a distinct variable constraint on x and z. (Contributed by Jim Kingdon, 6-Mar-2018.)
|- ( ph -> A. x ph )   &    |- ( ps -> A. z ps )   &    |- ( z = y -> ( ph <-> ps ) )   =>    |- ( -. A. x x = y -> ( ps -> A. x ps ) )
Theorem	hbsb4 2005	A variable not free remains so after substitution with a distinct variable. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
|- ( ph -> A. z ph )   =>    |- ( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) )
Theorem	hbsb4t 2006	A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2005). (Contributed by NM, 7-Apr-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( A. x A. z ( ph -> A. z ph ) -> ( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) ) )
Theorem	nfsb4t 2007	A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2005). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof rewritten by Jim Kingdon, 9-May-2018.)
|- ( A. x F/ z ph -> ( -. A. z z = y -> F/ z [ y / x ] ph ) )
Theorem	dvelimf 2008	Version of dvelim 2010 without any variable restrictions. (Contributed by NM, 1-Oct-2002.)
|- ( ph -> A. x ph )   &    |- ( ps -> A. z ps )   &    |- ( z = y -> ( ph <-> ps ) )   =>    |- ( -. A. x x = y -> ( ps -> A. x ps ) )
Theorem	dvelimdf 2009	Deduction form of dvelimf 2008. This version may be useful if we want to avoid ax-17 1519 and use ax-16 1807 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
|- F/ x ph   &    |- F/ z ph   &    |- ( ph -> F/ x ps )   &    |- ( ph -> F/ z ch )   &    |- ( ph -> ( z = y -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( -. A. x x = y -> F/ x ch ) )
Theorem	dvelim 2010*	This theorem can be used to eliminate a distinct variable restriction on x and z and replace it with the "distinctor" -. A. x x = y as an antecedent. ph normally has z free and can be read ph ( z ), and ps substitutes y for z and can be read ph ( y ). We don't require that x and y be distinct: if they aren't, the distinctor will become false (in multiple-element domains of discourse) and "protect" the consequent. To obtain a closed-theorem form of this inference, prefix the hypotheses with A. x A. z, conjoin them, and apply dvelimdf 2009. Other variants of this theorem are dvelimf 2008 (with no distinct variable restrictions) and dvelimALT 2003 (that avoids ax-10 1498). (Contributed by NM, 23-Nov-1994.)
|- ( ph -> A. x ph )   &    |- ( z = y -> ( ph <-> ps ) )   =>    |- ( -. A. x x = y -> ( ps -> A. x ps ) )
Theorem	dvelimor 2011*	Disjunctive distinct variable constraint elimination. A user of this theorem starts with a formula ph (containing z) and a distinct variable constraint between x and z. The theorem makes it possible to replace the distinct variable constraint with the disjunct A. x x = y ( ps is just a version of ph with y substituted for z). (Contributed by Jim Kingdon, 11-May-2018.)
|- F/ x ph   &    |- ( z = y -> ( ph <-> ps ) )   =>    |- ( A. x x = y \/ F/ x ps )
Theorem	dveeq1 2012*	Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 19-Feb-2018.)
|- ( -. A. x x = y -> ( y = z -> A. x y = z ) )
Theorem	sbal2 2013*	Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.)
|- ( -. A. x x = y -> ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) )
Theorem	nfsb4or 2014	A variable not free remains so after substitution with a distinct variable. (Contributed by Jim Kingdon, 11-May-2018.)
|- F/ z ph   =>    |- ( A. z z = y \/ F/ z [ y / x ] ph )
Theorem	nfd2 2015	Deduce that x is not free in ps in a context. (Contributed by Wolf Lammen, 16-Sep-2021.)
|- ( ph -> ( E. x ps -> A. x ps ) )   =>    |- ( ph -> F/ x ps )
Theorem	hbe1a 2016	Dual statement of hbe1 1488. (Contributed by Wolf Lammen, 15-Sep-2021.)
|- ( E. x A. x ph -> A. x ph )
Theorem	nf5-1 2017	One direction of nf5 . (Contributed by Wolf Lammen, 16-Sep-2021.)
|- ( A. x ( ph -> A. x ph ) -> F/ x ph )
Theorem	nf5d 2018	Deduce that x is not free in ps in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> F/ x ps )
1.4.6  Existential uniqueness
Syntax	weu 2019	Extend wff definition to include existential uniqueness ("there exists a unique x such that ph").
wff E! x ph
Syntax	wmo 2020	Extend wff definition to include uniqueness ("there exists at most one x such that ph").
wff E* x ph
Theorem	eujust 2021*	A soundness justification theorem for df-eu 2022, showing that the definition is equivalent to itself with its dummy variable renamed. Note that y and z needn't be distinct variables. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( E. y A. x ( ph <-> x = y ) <-> E. z A. x ( ph <-> x = z ) )
Definition	df-eu 2022*	Define existential uniqueness, i.e., "there exists exactly one x such that ph". Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 2044, eu2 2063, eu3 2065, and eu5 2066 (which in some cases we show with a hypothesis ph -> A. y ph in place of a distinct variable condition on y and ph). Double uniqueness is tricky: E! x E! y ph does not mean "exactly one x and one y " (see 2eu4 2112). (Contributed by NM, 5-Aug-1993.)
|- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
Definition	df-mo 2023	Define "there exists at most one x such that ph". Here we define it in terms of existential uniqueness. Notation of [BellMachover] p. 460, whose definition we show as mo3 2073. For another possible definition see mo4 2080. (Contributed by NM, 5-Aug-1993.)
|- ( E* x ph <-> ( E. x ph -> E! x ph ) )
Theorem	euf 2024*	A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)
|- ( ph -> A. y ph )   =>    |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
Theorem	eubidh 2025	Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E! x ps <-> E! x ch ) )
Theorem	eubid 2026	Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)
|- F/ x ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E! x ps <-> E! x ch ) )
Theorem	eubidv 2027*	Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E! x ps <-> E! x ch ) )
Theorem	eubii 2028	Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
|- ( ph <-> ps )   =>    |- ( E! x ph <-> E! x ps )
Theorem	hbeu1 2029	Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.)
|- ( E! x ph -> A. x E! x ph )
Theorem	nfeu1 2030	Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/ x E! x ph
Theorem	nfmo1 2031	Bound-variable hypothesis builder for "at most one". (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/ x E* x ph
Theorem	sb8eu 2032	Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/ y ph   =>    |- ( E! x ph <-> E! y [ y / x ] ph )
Theorem	sb8mo 2033	Variable substitution for "at most one". (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- F/ y ph   =>    |- ( E* x ph <-> E* y [ y / x ] ph )
Theorem	nfeudv 2034*	Deduction version of nfeu 2038. Similar to nfeud 2035 but has the additional constraint that x and y must be distinct. (Contributed by Jim Kingdon, 25-May-2018.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x E! y ps )
Theorem	nfeud 2035	Deduction version of nfeu 2038. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 25-May-2018.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x E! y ps )
Theorem	nfmod 2036	Bound-variable hypothesis builder for "at most one". (Contributed by Mario Carneiro, 14-Nov-2016.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x E* y ps )
Theorem	nfeuv 2037*	Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 2038 but has the additional condition that x and y must be distinct. (Contributed by Jim Kingdon, 23-May-2018.)
|- F/ x ph   =>    |- F/ x E! y ph
Theorem	nfeu 2038	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.)
|- F/ x ph   =>    |- F/ x E! y ph
Theorem	nfmo 2039	Bound-variable hypothesis builder for "at most one". (Contributed by NM, 9-Mar-1995.)
|- F/ x ph   =>    |- F/ x E* y ph
Theorem	hbeu 2040	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.)
|- ( ph -> A. x ph )   =>    |- ( E! y ph -> A. x E! y ph )
Theorem	hbeud 2041	Deduction version of hbeu 2040. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 25-May-2018.)
|- ( ph -> A. x ph )   &    |- ( ph -> A. y ph )   &    |- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> ( E! y ps -> A. x E! y ps ) )
Theorem	sb8euh 2042	Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.)
|- ( ph -> A. y ph )   =>    |- ( E! x ph <-> E! y [ y / x ] ph )
Theorem	cbveu 2043	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x ph <-> E! y ps )
Theorem	eu1 2044*	An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.)
|- ( ph -> A. y ph )   =>    |- ( E! x ph <-> E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) )
Theorem	euor 2045	Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 21-Oct-2005.)
|- ( ph -> A. x ph )   =>    |- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) )
Theorem	euorv 2046*	Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 23-Mar-1995.)
|- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) )
Theorem	mo2n 2047*	There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.)
|- F/ y ph   =>    |- ( -. E. x ph -> E. y A. x ( ph -> x = y ) )
Theorem	mon 2048	There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.)
|- ( -. E. x ph -> E* x ph )
Theorem	euex 2049	Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( E! x ph -> E. x ph )
Theorem	eumo0 2050*	Existential uniqueness implies "at most one". (Contributed by NM, 8-Jul-1994.)
|- ( ph -> A. y ph )   =>    |- ( E! x ph -> E. y A. x ( ph -> x = y ) )
Theorem	eumo 2051	Existential uniqueness implies "at most one". (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)
|- ( E! x ph -> E* x ph )
Theorem	eumoi 2052	"At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
|- E! x ph   =>    |- E* x ph
Theorem	mobidh 2053	Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E* x ps <-> E* x ch ) )
Theorem	mobid 2054	Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.)
|- F/ x ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E* x ps <-> E* x ch ) )
Theorem	mobidv 2055*	Formula-building rule for "at most one" quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E* x ps <-> E* x ch ) )
Theorem	mobii 2056	Formula-building rule for "at most one" quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)
|- ( ps <-> ch )   =>    |- ( E* x ps <-> E* x ch )
Theorem	hbmo1 2057	Bound-variable hypothesis builder for "at most one". (Contributed by NM, 8-Mar-1995.)
|- ( E* x ph -> A. x E* x ph )
Theorem	hbmo 2058	Bound-variable hypothesis builder for "at most one". (Contributed by NM, 9-Mar-1995.)
|- ( ph -> A. x ph )   =>    |- ( E* y ph -> A. x E* y ph )
Theorem	cbvmo 2059	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E* x ph <-> E* y ps )
Theorem	mo23 2060*	An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.)
|- F/ y ph   =>    |- ( E. y A. x ( ph -> x = y ) -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
Theorem	mor 2061*	Converse of mo23 2060 with an additional E. x ph condition. (Contributed by Jim Kingdon, 25-Jun-2018.)
|- F/ y ph   =>    |- ( E. x ph -> ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> E. y A. x ( ph -> x = y ) ) )
Theorem	modc 2062*	Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.)
|- F/ y ph   =>    |- (DECID E. x ph -> ( E. y A. x ( ph -> x = y ) <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )
Theorem	eu2 2063*	An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.)
|- F/ y ph   =>    |- ( E! x ph <-> ( E. x ph /\ A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )
Theorem	eu3h 2064*	An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.)
|- ( ph -> A. y ph )   =>    |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
Theorem	eu3 2065*	An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)
|- F/ y ph   =>    |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
Theorem	eu5 2066	Uniqueness in terms of "at most one". (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)
|- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
Theorem	exmoeu2 2067	Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)
|- ( E. x ph -> ( E* x ph <-> E! x ph ) )
Theorem	moabs 2068	Absorption of existence condition by "at most one". (Contributed by NM, 4-Nov-2002.)
|- ( E* x ph <-> ( E. x ph -> E* x ph ) )
Theorem	exmodc 2069	If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.)
|- (DECID E. x ph -> ( E. x ph \/ E* x ph ) )
Theorem	exmonim 2070	There is at most one of something which does not exist. Unlike exmodc 2069 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.)
|- ( -. E. x ph -> E* x ph )
Theorem	mo2r 2071*	A condition which implies "at most one". (Contributed by Jim Kingdon, 2-Jul-2018.)
|- F/ y ph   =>    |- ( E. y A. x ( ph -> x = y ) -> E* x ph )
Theorem	mo3h 2072*	Alternate definition of "at most one". Definition of [BellMachover] p. 460, except that definition has the side condition that y not occur in ph in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (New usage is discouraged.)
|- ( ph -> A. y ph )   =>    |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
Theorem	mo3 2073*	Alternate definition of "at most one". Definition of [BellMachover] p. 460, except that definition has the side condition that y not occur in ph in place of our hypothesis. (Contributed by NM, 8-Mar-1995.)
|- F/ y ph   =>    |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
Theorem	mo2dc 2074*	Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.)
|- F/ y ph   =>    |- (DECID E. x ph -> ( E* x ph <-> E. y A. x ( ph -> x = y ) ) )
Theorem	euan 2075	Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( ph -> A. x ph )   =>    |- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) )
Theorem	euanv 2076*	Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.)
|- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) )
Theorem	euor2 2077	Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( -. E. x ph -> ( E! x ( ph \/ ps ) <-> E! x ps ) )
Theorem	sbmo 2078*	Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( [ y / x ] E* z ph <-> E* z [ y / x ] ph )
Theorem	mo4f 2079*	"At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)
|- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )
Theorem	mo4 2080*	"At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )
Theorem	eu4 2081*	Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x ph <-> ( E. x ph /\ A. x A. y ( ( ph /\ ps ) -> x = y ) ) )
Theorem	exmoeudc 2082	Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.)
|- (DECID E. x ph -> ( E. x ph <-> ( E* x ph -> E! x ph ) ) )
Theorem	moim 2083	"At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)
|- ( A. x ( ph -> ps ) -> ( E* x ps -> E* x ph ) )
Theorem	moimi 2084	"At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.)
|- ( ph -> ps )   =>    |- ( E* x ps -> E* x ph )
Theorem	moimv 2085*	Move antecedent outside of "at most one". (Contributed by NM, 28-Jul-1995.)
|- ( E* x ( ph -> ps ) -> ( ph -> E* x ps ) )
Theorem	euimmo 2086	Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.)
|- ( A. x ( ph -> ps ) -> ( E! x ps -> E* x ph ) )
Theorem	euim 2087	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.)
|- ( ( E. x ph /\ A. x ( ph -> ps ) ) -> ( E! x ps -> E! x ph ) )
Theorem	moan 2088	"At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)
|- ( E* x ph -> E* x ( ps /\ ph ) )
Theorem	moani 2089	"At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)
|- E* x ph   =>    |- E* x ( ps /\ ph )
Theorem	moor 2090	"At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)
|- ( E* x ( ph \/ ps ) -> E* x ph )
Theorem	mooran1 2091	"At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( ( E* x ph \/ E* x ps ) -> E* x ( ph /\ ps ) )
Theorem	mooran2 2092	"At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( E* x ( ph \/ ps ) -> ( E* x ph /\ E* x ps ) )
Theorem	moanim 2093	Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 3-Dec-2001.)
|- F/ x ph   =>    |- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) )
Theorem	moanimv 2094*	Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 23-Mar-1995.)
|- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) )
Theorem	moaneu 2095	Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.)
|- E* x ( ph /\ E! x ph )
Theorem	moanmo 2096	Nested at-most-one quantifiers. (Contributed by NM, 25-Jan-2006.)
|- E* x ( ph /\ E* x ph )
Theorem	mopick 2097	"At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.)
|- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
Theorem	eupick 2098	Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing x such that ph is true, and there is also an x (actually the same one) such that ph and ps are both true, then ph implies ps 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.)
|- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
Theorem	eupicka 2099	Version of eupick 2098 with closed formulas. (Contributed by NM, 6-Sep-2008.)
|- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> A. x ( ph -> ps ) )
Theorem	eupickb 2100	Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)
|- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( ph <-> ps ) )