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Theorem List for Intuitionistic Logic Explorer - 1901-2000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem2exsb 1901* An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.)

TheoremdvelimALT 1902* Version of dvelim 1909 that doesn't use ax-10 1412. Because it has different distinct variable constraints than dvelim 1909 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.)

Theoremdvelimfv 1903* Like dvelimf 1907 but with a distinct variable constraint on and . (Contributed by Jim Kingdon, 6-Mar-2018.)

Theoremhbsb4 1904 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.)

Theoremhbsb4t 1905 A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1904). (Contributed by NM, 7-Apr-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnfsb4t 1906 A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1904). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof rewritten by Jim Kingdon, 9-May-2018.)

Theoremdvelimf 1907 Version of dvelim 1909 without any variable restrictions. (Contributed by NM, 1-Oct-2002.)

Theoremdvelimdf 1908 Deduction form of dvelimf 1907. This version may be useful if we want to avoid ax-17 1435 and use ax-16 1711 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)

Theoremdvelim 1909* This theorem can be used to eliminate a distinct variable restriction on and and replace it with the "distinctor" as an antecedent. normally has free and can be read , and substitutes for and can be read . We don't require that and 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 , conjoin them, and apply dvelimdf 1908.

Other variants of this theorem are dvelimf 1907 (with no distinct variable restrictions) and dvelimALT 1902 (that avoids ax-10 1412). (Contributed by NM, 23-Nov-1994.)

Theoremdvelimor 1910* Disjunctive distinct variable constraint elimination. A user of this theorem starts with a formula (containing ) and a distinct variable constraint between and . The theorem makes it possible to replace the distinct variable constraint with the disjunct ( is just a version of with substituted for ). (Contributed by Jim Kingdon, 11-May-2018.)

Theoremdveeq1 1911* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 19-Feb-2018.)

Theoremdveel1 1912* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)

Theoremdveel2 1913* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)

Theoremsbal2 1914* Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.)

Theoremnfsb4or 1915 A variable not free remains so after substitution with a distinct variable. (Contributed by Jim Kingdon, 11-May-2018.)

1.4.6  Existential uniqueness

Syntaxweu 1916 Extend wff definition to include existential uniqueness ("there exists a unique such that ").

Syntaxwmo 1917 Extend wff definition to include uniqueness ("there exists at most one such that ").

Theoremeujust 1918* A soundness justification theorem for df-eu 1919, 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 1919* 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 1941, eu2 1960, eu3 1962, and eu5 1963 (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 2009). (Contributed by NM, 5-Aug-1993.)

Definitiondf-mo 1920 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 1970. For another possible definition see mo4 1977. (Contributed by NM, 5-Aug-1993.)

Theoremeuf 1921* A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)

Theoremeubidh 1922 Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Theoremeubid 1923 Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Theoremeubidv 1924* Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Theoremeubii 1925 Introduce uniqueness quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremhbeu1 1926 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.)

Theoremnfeu1 1927 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfmo1 1928 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremsb8eu 1929 Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremsb8mo 1930 Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremnfeudv 1931* Deduction version of nfeu 1935. Similar to nfeud 1932 but has the additional constraint that and must be distinct. (Contributed by Jim Kingdon, 25-May-2018.)

Theoremnfeud 1932 Deduction version of nfeu 1935. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 25-May-2018.)

Theoremnfmod 1933 Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremnfeuv 1934* Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 1935 but has the additional constraint that and must be distinct. (Contributed by Jim Kingdon, 23-May-2018.)

Theoremnfeu 1935 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 1936 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)

Theoremhbeu 1937 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 1938 Deduction version of hbeu 1937. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 25-May-2018.)

Theoremsb8euh 1939 Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.)

Theoremcbveu 1940 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremeu1 1941* An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.)

Theoremeuor 1942 Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.)

Theoremeuorv 1943* Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremmo2n 1944* There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.)

Theoremmon 1945 There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.)

Theoremeuex 1946 Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremeumo0 1947* Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)

Theoremeumo 1948 Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)

Theoremeumoi 1949 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)

Theoremmobidh 1950 Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)

Theoremmobid 1951 Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)

Theoremmobidv 1952* Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.)

Theoremmobii 1953 Formula-building rule for "at most one" quantifier (inference rule). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)

Theoremhbmo1 1954 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.)

Theoremhbmo 1955 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)

Theoremcbvmo 1956 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)

Theoremmo23 1957* An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.)

Theoremmor 1958* Converse of mo23 1957 with an additional condition. (Contributed by Jim Kingdon, 25-Jun-2018.)

Theoremmodc 1959* Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.)
DECID

Theoremeu2 1960* An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.)

Theoremeu3h 1961* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.)

Theoremeu3 1962* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)

Theoremeu5 1963 Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)

Theoremexmoeu2 1964 Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)

Theoremmoabs 1965 Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)

Theoremexmodc 1966 If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.)
DECID

Theoremexmonim 1967 There is at most one of something which does not exist. Unlike exmodc 1966 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.)

Theoremmo2r 1968* A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.)

Theoremmo3h 1969* 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 1970* 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 1971* Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.)
DECID

Theoremeuan 1972 Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremeuanv 1973* Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremeuor2 1974 Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremsbmo 1975* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremmo4f 1976* "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)

Theoremmo4 1977* "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Theoremeu4 1978* Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Theoremexmoeudc 1979 Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.)
DECID

Theoremmoim 1980 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)

Theoremmoimi 1981 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.)

Theoremmoimv 1982* Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)

Theoremeuimmo 1983 Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.)

Theoremeuim 1984 Add existential uniqueness 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 1985 "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)

Theoremmoani 1986 "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)

Theoremmoor 1987 "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)

Theoremmooran1 1988 "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmooran2 1989 "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmoanim 1990 Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.)

Theoremmoanimv 1991* Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremmoaneu 1992 Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.)

Theoremmoanmo 1993 Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.)

Theoremmopick 1994 "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.)

Theoremeupick 1995 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 1996 Version of eupick 1995 with closed formulas. (Contributed by NM, 6-Sep-2008.)

Theoremeupickb 1997 Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)

Theoremeupickbi 1998 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremmopick2 1999 "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 1538. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmoexexdc 2000 "At most one" double quantification. (Contributed by Jim Kingdon, 5-Jul-2018.)
DECID

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