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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sb9i 1901 | Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.) |
Theorem | sbnf2 1902* | Two ways of expressing " is (effectively) not free in ." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | hbsbd 1903* | Deduction version of hbsb 1868. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.) |
Theorem | 2sb5 1904* | Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
Theorem | 2sb6 1905* | Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
Theorem | sbcom2v 1906* | Lemma for proving sbcom2 1908. It is the same as sbcom2 1908 but with additional distinct variable constraints on and , and on and . (Contributed by Jim Kingdon, 19-Feb-2018.) |
Theorem | sbcom2v2 1907* | Lemma for proving sbcom2 1908. It is the same as sbcom2v 1906 but removes the distinct variable constraint on and . (Contributed by Jim Kingdon, 19-Feb-2018.) |
Theorem | sbcom2 1908* | Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.) (Proof modified to be intuitionistic by Jim Kingdon, 19-Feb-2018.) |
Theorem | sb6a 1909* | Equivalence for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | 2sb5rf 1910* | Reversed double substitution. (Contributed by NM, 3-Feb-2005.) |
Theorem | 2sb6rf 1911* | Reversed double substitution. (Contributed by NM, 3-Feb-2005.) |
Theorem | dfsb7 1912* | An alternate definition of proper substitution df-sb 1690. By introducing a dummy variable in the definiens, we are able to eliminate any distinct variable restrictions among the variables , , and of the definiendum. No distinct variable conflicts arise because effectively insulates from . To achieve this, we use a chain of two substitutions in the form of sb5 1812, first for then for . Compare Definition 2.1'' of [Quine] p. 17. Theorem sb7f 1913 provides a version where and don't have to be distinct. (Contributed by NM, 28-Jan-2004.) |
Theorem | sb7f 1913* | This version of dfsb7 1912 does not require that and be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1462 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1690 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sb7af 1914* | An alternate definition of proper substitution df-sb 1690. Similar to dfsb7a 1915 but does not require that and be distinct. Similar to sb7f 1913 in that it involves a dummy variable , but expressed in terms of rather than . (Contributed by Jim Kingdon, 5-Feb-2018.) |
Theorem | dfsb7a 1915* | An alternate definition of proper substitution df-sb 1690. Similar to dfsb7 1912 in that it involves a dummy variable , but expressed in terms of rather than . For a version which only requires rather than and being distinct, see sb7af 1914. (Contributed by Jim Kingdon, 5-Feb-2018.) |
Theorem | sb10f 1916* | Hao Wang's identity axiom P6 in Irving Copi, Symbolic Logic (5th ed., 1979), p. 328. In traditional predicate calculus, this is a sole axiom for identity from which the usual ones can be derived. (Contributed by NM, 9-May-2005.) |
Theorem | sbid2v 1917* | An identity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.) |
Theorem | sbelx 1918* | Elimination of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbel2x 1919* | Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbalyz 1920* | Move universal quantifier in and out of substitution. Identical to sbal 1921 except that it has an additional distinct variable constraint on and . (Contributed by Jim Kingdon, 29-Dec-2017.) |
Theorem | sbal 1921* | Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.) |
Theorem | sbal1yz 1922* | Lemma for proving sbal1 1923. Same as sbal1 1923 but with an additional distinct variable constraint on and . (Contributed by Jim Kingdon, 23-Feb-2018.) |
Theorem | sbal1 1923* | A theorem used in elimination of disjoint variable restriction on and by replacing it with a distinctor . (Contributed by NM, 5-Aug-1993.) (Proof rewitten by Jim Kingdon, 24-Feb-2018.) |
Theorem | sbexyz 1924* | Move existential quantifier in and out of substitution. Identical to sbex 1925 except that it has an additional distinct variable constraint on and . (Contributed by Jim Kingdon, 29-Dec-2017.) |
Theorem | sbex 1925* | Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.) |
Theorem | sbalv 1926* | Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.) |
Theorem | sbco4lem 1927* | Lemma for sbco4 1928. It replaces the temporary variable with another temporary variable . (Contributed by Jim Kingdon, 26-Sep-2018.) |
Theorem | sbco4 1928* | Two ways of exchanging two variables. Both sides of the biconditional exchange and , either via two temporary variables and , or a single temporary . (Contributed by Jim Kingdon, 25-Sep-2018.) |
Theorem | exsb 1929* | An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.) |
Theorem | 2exsb 1930* | An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) |
Theorem | dvelimALT 1931* | Version of dvelim 1938 that doesn't use ax-10 1439. Because it has different distinct variable constraints than dvelim 1938 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.) |
Theorem | dvelimfv 1932* | Like dvelimf 1936 but with a distinct variable constraint on and . (Contributed by Jim Kingdon, 6-Mar-2018.) |
Theorem | hbsb4 1933 | 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.) |
Theorem | hbsb4t 1934 | A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1933). (Contributed by NM, 7-Apr-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | nfsb4t 1935 | A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1933). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof rewritten by Jim Kingdon, 9-May-2018.) |
Theorem | dvelimf 1936 | Version of dvelim 1938 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) |
Theorem | dvelimdf 1937 | Deduction form of dvelimf 1936. This version may be useful if we want to avoid ax-17 1462 and use ax-16 1739 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) |
Theorem | dvelim 1938* |
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 1937. Other variants of this theorem are dvelimf 1936 (with no distinct variable restrictions) and dvelimALT 1931 (that avoids ax-10 1439). (Contributed by NM, 23-Nov-1994.) |
Theorem | dvelimor 1939* | 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.) |
Theorem | dveeq1 1940* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 19-Feb-2018.) |
Theorem | dveel1 1941* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) |
Theorem | dveel2 1942* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) |
Theorem | sbal2 1943* | Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) |
Theorem | nfsb4or 1944 | A variable not free remains so after substitution with a distinct variable. (Contributed by Jim Kingdon, 11-May-2018.) |
Syntax | weu 1945 | Extend wff definition to include existential uniqueness ("there exists a unique such that "). |
Syntax | wmo 1946 | Extend wff definition to include uniqueness ("there exists at most one such that "). |
Theorem | eujust 1947* | A soundness justification theorem for df-eu 1948, 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.) |
Definition | df-eu 1948* | 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 1970, eu2 1989, eu3 1991, and eu5 1992 (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 2038). (Contributed by NM, 5-Aug-1993.) |
Definition | df-mo 1949 | 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 1999. For another possible definition see mo4 2006. (Contributed by NM, 5-Aug-1993.) |
Theorem | euf 1950* | A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.) |
Theorem | eubidh 1951 | Formula-building rule for unique existential quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.) |
Theorem | eubid 1952 | Formula-building rule for unique existential quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.) |
Theorem | eubidv 1953* | Formula-building rule for unique existential quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.) |
Theorem | eubii 1954 | Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | hbeu1 1955 | Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) |
Theorem | nfeu1 1956 | Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Theorem | nfmo1 1957 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Theorem | sb8eu 1958 | Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Theorem | sb8mo 1959 | Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | nfeudv 1960* | Deduction version of nfeu 1964. Similar to nfeud 1961 but has the additional constraint that and must be distinct. (Contributed by Jim Kingdon, 25-May-2018.) |
Theorem | nfeud 1961 | Deduction version of nfeu 1964. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 25-May-2018.) |
Theorem | nfmod 1962 | Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.) |
Theorem | nfeuv 1963* | Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 1964 but has the additional constraint that and must be distinct. (Contributed by Jim Kingdon, 23-May-2018.) |
Theorem | nfeu 1964 | 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.) |
Theorem | nfmo 1965 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.) |
Theorem | hbeu 1966 | 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.) |
Theorem | hbeud 1967 | Deduction version of hbeu 1966. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 25-May-2018.) |
Theorem | sb8euh 1968 | Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.) |
Theorem | cbveu 1969 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Theorem | eu1 1970* | An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) |
Theorem | euor 1971 | Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) |
Theorem | euorv 1972* | Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 23-Mar-1995.) |
Theorem | mo2n 1973* | There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.) |
Theorem | mon 1974 | There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.) |
Theorem | euex 1975 | Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Theorem | eumo0 1976* | Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.) |
Theorem | eumo 1977 | Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.) |
Theorem | eumoi 1978 | "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
Theorem | mobidh 1979 | Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.) |
Theorem | mobid 1980 | Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.) |
Theorem | mobidv 1981* | Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.) |
Theorem | mobii 1982 | Formula-building rule for "at most one" quantifier (inference rule). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.) |
Theorem | hbmo1 1983 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) |
Theorem | hbmo 1984 | Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.) |
Theorem | cbvmo 1985 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) |
Theorem | mo23 1986* | An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.) |
Theorem | mor 1987* | Converse of mo23 1986 with an additional condition. (Contributed by Jim Kingdon, 25-Jun-2018.) |
Theorem | modc 1988* | Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.) |
DECID | ||
Theorem | eu2 1989* | An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) |
Theorem | eu3h 1990* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.) |
Theorem | eu3 1991* | An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) |
Theorem | eu5 1992 | Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.) |
Theorem | exmoeu2 1993 | Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.) |
Theorem | moabs 1994 | Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.) |
Theorem | exmodc 1995 | If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.) |
DECID | ||
Theorem | exmonim 1996 | There is at most one of something which does not exist. Unlike exmodc 1995 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.) |
Theorem | mo2r 1997* | A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.) |
Theorem | mo3h 1998* | 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.) |
Theorem | mo3 1999* | 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.) |
Theorem | mo2dc 2000* | Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.) |
DECID |
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