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

Theoremnfsb4or 2001 A variable not free remains so after substitution with a distinct variable. (Contributed by Jim Kingdon, 11-May-2018.)

Theoremnfd2 2002 Deduce that is not free in in a context. (Contributed by Wolf Lammen, 16-Sep-2021.)

Theoremhbe1a 2003 Dual statement of hbe1 1475. (Contributed by Wolf Lammen, 15-Sep-2021.)

Theoremnf5-1 2004 One direction of nf5 . (Contributed by Wolf Lammen, 16-Sep-2021.)

Theoremnf5d 2005 Deduce that is not free in in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)

1.4.6  Existential uniqueness

Syntaxweu 2006 Extend wff definition to include existential uniqueness ("there exists a unique such that ").

Syntaxwmo 2007 Extend wff definition to include uniqueness ("there exists at most one such that ").

Theoremeujust 2008* A soundness justification theorem for df-eu 2009, showing that the definition is equivalent to itself with its dummy variable renamed. Note that and needn't be distinct variables. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Definitiondf-eu 2009* Define existential uniqueness, i.e. "there exists exactly one such that ." Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 2031, eu2 2050, eu3 2052, and eu5 2053 (which in some cases we show with a hypothesis in place of a distinct variable condition on and ). Double uniqueness is tricky: does not mean "exactly one and one " (see 2eu4 2099). (Contributed by NM, 5-Aug-1993.)

Definitiondf-mo 2010 Define "there exists at most one such that ." Here we define it in terms of existential uniqueness. Notation of [BellMachover] p. 460, whose definition we show as mo3 2060. For another possible definition see mo4 2067. (Contributed by NM, 5-Aug-1993.)

Theoremeuf 2011* A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)

Theoremeubidh 2012 Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)

Theoremeubid 2013 Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)

Theoremeubidv 2014* Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.)

Theoremeubii 2015 Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremhbeu1 2016 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.)

Theoremnfeu1 2017 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfmo1 2018 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremsb8eu 2019 Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremsb8mo 2020 Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremnfeudv 2021* Deduction version of nfeu 2025. Similar to nfeud 2022 but has the additional constraint that and must be distinct. (Contributed by Jim Kingdon, 25-May-2018.)

Theoremnfeud 2022 Deduction version of nfeu 2025. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 25-May-2018.)

Theoremnfmod 2023 Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremnfeuv 2024* Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 2025 but has the additional constraint that and must be distinct. (Contributed by Jim Kingdon, 23-May-2018.)

Theoremnfeu 2025 Bound-variable hypothesis builder for existential uniqueness. Note that and 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.)

Theoremnfmo 2026 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)

Theoremhbeu 2027 Bound-variable hypothesis builder for uniqueness. Note that and needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Proof rewritten by Jim Kingdon, 24-May-2018.)

Theoremhbeud 2028 Deduction version of hbeu 2027. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 25-May-2018.)

Theoremsb8euh 2029 Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.)

Theoremcbveu 2030 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremeu1 2031* An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.)

Theoremeuor 2032 Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 21-Oct-2005.)

Theoremeuorv 2033* Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremmo2n 2034* There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.)

Theoremmon 2035 There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.)

Theoremeuex 2036 Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremeumo0 2037* Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)

Theoremeumo 2038 Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)

Theoremeumoi 2039 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)

Theoremmobidh 2040 Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.)

Theoremmobid 2041 Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.)

Theoremmobidv 2042* Formula-building rule for "at most one" quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016.)

Theoremmobii 2043 Formula-building rule for "at most one" quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)

Theoremhbmo1 2044 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.)

Theoremhbmo 2045 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)

Theoremcbvmo 2046 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)

Theoremmo23 2047* An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.)

Theoremmor 2048* Converse of mo23 2047 with an additional condition. (Contributed by Jim Kingdon, 25-Jun-2018.)

Theoremmodc 2049* Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.)
DECID

Theoremeu2 2050* An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.)

Theoremeu3h 2051* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.)

Theoremeu3 2052* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)

Theoremeu5 2053 Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)

Theoremexmoeu2 2054 Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)

Theoremmoabs 2055 Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)

Theoremexmodc 2056 If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.)
DECID

Theoremexmonim 2057 There is at most one of something which does not exist. Unlike exmodc 2056 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.)

Theoremmo2r 2058* A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.)

Theoremmo3h 2059* Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that not occur in in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (New usage is discouraged.)

Theoremmo3 2060* Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that not occur in in place of our hypothesis. (Contributed by NM, 8-Mar-1995.)

Theoremmo2dc 2061* Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.)
DECID

Theoremeuan 2062 Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremeuanv 2063* Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremeuor2 2064 Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremsbmo 2065* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremmo4f 2066* "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)

Theoremmo4 2067* "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Theoremeu4 2068* Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Theoremexmoeudc 2069 Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.)
DECID

Theoremmoim 2070 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)

Theoremmoimi 2071 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.)

Theoremmoimv 2072* Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)

Theoremeuimmo 2073 Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.)

Theoremeuim 2074 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.)

Theoremmoan 2075 "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)

Theoremmoani 2076 "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)

Theoremmoor 2077 "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)

Theoremmooran1 2078 "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmooran2 2079 "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmoanim 2080 Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 3-Dec-2001.)

Theoremmoanimv 2081* Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremmoaneu 2082 Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.)

Theoremmoanmo 2083 Nested at-most-one quantifiers. (Contributed by NM, 25-Jan-2006.)

Theoremmopick 2084 "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.)

Theoremeupick 2085 Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing such that is true, and there is also an (actually the same one) such that and are both true, then implies regardless of . 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.)

Theoremeupicka 2086 Version of eupick 2085 with closed formulas. (Contributed by NM, 6-Sep-2008.)

Theoremeupickb 2087 Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)

Theoremeupickbi 2088 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremmopick2 2089 "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 1611. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmoexexdc 2090 "At most one" double quantification. (Contributed by Jim Kingdon, 5-Jul-2018.)
DECID

Theoremeuexex 2091 Existential uniqueness and "at most one" double quantification. (Contributed by Jim Kingdon, 28-Dec-2018.)

Theorem2moex 2092 Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)

Theorem2euex 2093 Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theorem2eumo 2094 Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.)

Theorem2eu2ex 2095 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)

Theorem2moswapdc 2096 A condition allowing swap of "at most one" and existential quantifiers. (Contributed by Jim Kingdon, 6-Jul-2018.)
DECID

Theorem2euswapdc 2097 A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Jim Kingdon, 7-Jul-2018.)
DECID

Theorem2exeu 2098 Double existential uniqueness implies double unique existential quantification. (Contributed by NM, 3-Dec-2001.)

Theorem2eu4 2099* This theorem provides us with a definition of double existential uniqueness ("exactly one and exactly one "). Naively one might think (incorrectly) that it could be defined by . See 2exeu 2098 for a one-way implication. (Contributed by NM, 3-Dec-2001.)

Theorem2eu7 2100 Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)

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