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	hbsb4t 2001	A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2000). (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 2002	A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2000). (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 2003	Version of dvelim 2005 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 2004	Deduction form of dvelimf 2003. This version may be useful if we want to avoid ax-17 1514 and use ax-16 1802 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 2005*	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 2004. Other variants of this theorem are dvelimf 2003 (with no distinct variable restrictions) and dvelimALT 1998 (that avoids ax-10 1493). (Contributed by NM, 23-Nov-1994.)
|- ( ph -> A. x ph )   &    |- ( z = y -> ( ph <-> ps ) )   =>    |- ( -. A. x x = y -> ( ps -> A. x ps ) )
Theorem	dvelimor 2006*	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 2007*	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 2008*	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 2009	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 2010	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 2011	Dual statement of hbe1 1483. (Contributed by Wolf Lammen, 15-Sep-2021.)
|- ( E. x A. x ph -> A. x ph )
Theorem	nf5-1 2012	One direction of nf5 . (Contributed by Wolf Lammen, 16-Sep-2021.)
|- ( A. x ( ph -> A. x ph ) -> F/ x ph )
Theorem	nf5d 2013	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 2014	Extend wff definition to include existential uniqueness ("there exists a unique x such that ph").
wff E! x ph
Syntax	wmo 2015	Extend wff definition to include uniqueness ("there exists at most one x such that ph").
wff E* x ph
Theorem	eujust 2016*	A soundness justification theorem for df-eu 2017, 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 2017*	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 2039, eu2 2058, eu3 2060, and eu5 2061 (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 2107). (Contributed by NM, 5-Aug-1993.)
|- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
Definition	df-mo 2018	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 2068. For another possible definition see mo4 2075. (Contributed by NM, 5-Aug-1993.)
|- ( E* x ph <-> ( E. x ph -> E! x ph ) )
Theorem	euf 2019*	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 2020	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 2021	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 2022*	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 2023	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 2024	Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.)
|- ( E! x ph -> A. x E! x ph )
Theorem	nfeu1 2025	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 2026	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 2027	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 2028	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 2029*	Deduction version of nfeu 2033. Similar to nfeud 2030 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 2030	Deduction version of nfeu 2033. (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 2031	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 2032*	Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 2033 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 2033	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 2034	Bound-variable hypothesis builder for "at most one". (Contributed by NM, 9-Mar-1995.)
|- F/ x ph   =>    |- F/ x E* y ph
Theorem	hbeu 2035	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 2036	Deduction version of hbeu 2035. (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 2037	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 2038	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 2039*	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 2040	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 2041*	Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 23-Mar-1995.)
|- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) )
Theorem	mo2n 2042*	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 2043	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 2044	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 2045*	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 2046	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 2047	"At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
|- E! x ph   =>    |- E* x ph
Theorem	mobidh 2048	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 2049	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 2050*	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 2051	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 2052	Bound-variable hypothesis builder for "at most one". (Contributed by NM, 8-Mar-1995.)
|- ( E* x ph -> A. x E* x ph )
Theorem	hbmo 2053	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 2054	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 2055*	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 2056*	Converse of mo23 2055 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 2057*	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 2058*	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 2059*	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 2060*	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 2061	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 2062	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 2063	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 2064	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 2065	There is at most one of something which does not exist. Unlike exmodc 2064 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.)
|- ( -. E. x ph -> E* x ph )
Theorem	mo2r 2066*	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 2067*	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 2068*	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 2069*	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 2070	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 2071*	Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.)
|- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) )
Theorem	euor2 2072	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 2073*	Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( [ y / x ] E* z ph <-> E* z [ y / x ] ph )
Theorem	mo4f 2074*	"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 2075*	"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 2076*	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 2077	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 2078	"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 2079	"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 2080*	Move antecedent outside of "at most one". (Contributed by NM, 28-Jul-1995.)
|- ( E* x ( ph -> ps ) -> ( ph -> E* x ps ) )
Theorem	euimmo 2081	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 2082	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 2083	"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 2084	"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 2085	"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 2086	"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 2087	"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 2088	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 2089*	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 2090	Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.)
|- E* x ( ph /\ E! x ph )
Theorem	moanmo 2091	Nested at-most-one quantifiers. (Contributed by NM, 25-Jan-2006.)
|- E* x ( ph /\ E* x ph )
Theorem	mopick 2092	"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 2093	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 2094	Version of eupick 2093 with closed formulas. (Contributed by NM, 6-Sep-2008.)
|- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> A. x ( ph -> ps ) )
Theorem	eupickb 2095	Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)
|- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( ph <-> ps ) )
Theorem	eupickbi 2096	Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( E! x ph -> ( E. x ( ph /\ ps ) <-> A. x ( ph -> ps ) ) )
Theorem	mopick2 2097	"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 1619. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( ( E* x ph /\ E. x ( ph /\ ps ) /\ E. x ( ph /\ ch ) ) -> E. x ( ph /\ ps /\ ch ) )
Theorem	moexexdc 2098	"At most one" double quantification. (Contributed by Jim Kingdon, 5-Jul-2018.)
|- F/ y ph   =>    |- (DECID E. x ph -> ( ( E* x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) ) )
Theorem	euexex 2099	Existential uniqueness and "at most one" double quantification. (Contributed by Jim Kingdon, 28-Dec-2018.)
|- F/ y ph   =>    |- ( ( E! x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) )
Theorem	2moex 2100	Double quantification with "at most one". (Contributed by NM, 3-Dec-2001.)
|- ( E* x E. y ph -> A. y E* x ph )