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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cbvaldva 1901* | Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Theorem | cbvexdva 1902* | Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Theorem | cbvex4v 1903* | Rule used to change bound variables, using implicit substitition. (Contributed by NM, 26-Jul-1995.) |
Theorem | eean 1904 | Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | eeanv 1905* | Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) |
Theorem | eeeanv 1906* | Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | ee4anv 1907* | Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.) |
Theorem | ee8anv 1908* | Rearrange existential quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.) |
Theorem | nexdv 1909* | Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.) |
Theorem | chvarv 1910* | Implicit substitution of for into a theorem. (Contributed by NM, 20-Apr-1994.) |
Theorem | cleljust 1911* | When the class variables of set theory are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 1482 with the class variables in wcel 1481. (Contributed by NM, 28-Jan-2004.) |
Theorem | hbs1 1912* | is not free in when and are distinct. (Contributed by NM, 5-Aug-1993.) (Proof by Jim Kingdon, 16-Dec-2017.) (New usage is discouraged.) |
Theorem | nfs1v 1913* | is not free in when and are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | sbhb 1914* | Two ways of expressing " is (effectively) not free in ." (Contributed by NM, 29-May-2009.) |
Theorem | hbsbv 1915* | This is a version of hbsb 1923 with an extra distinct variable constraint, on and . (Contributed by Jim Kingdon, 25-Dec-2017.) |
Theorem | nfsbxy 1916* | Similar to hbsb 1923 but with an extra distinct variable constraint, on and . (Contributed by Jim Kingdon, 19-Mar-2018.) |
Theorem | nfsbxyt 1917* | Closed form of nfsbxy 1916. (Contributed by Jim Kingdon, 9-May-2018.) |
Theorem | sbco2vlem 1918* | This is a version of sbco2 1939 where is distinct from and from . It is a lemma on the way to proving sbco2v 1922 which only requires that and be distinct. (Contributed by Jim Kingdon, 25-Dec-2017.) Remove one disjoint variable condition. (Revised by Jim Kingdon, 3-Feb-2018.) |
Theorem | sbco2vh 1919* | This is a version of sbco2 1939 where is distinct from . (Contributed by Jim Kingdon, 12-Feb-2018.) |
Theorem | nfsb 1920* | If is not free in , it is not free in when and are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.) |
Theorem | nfsbv 1921* | If is not free in , it is not free in when is distinct from and . Version of nfsb 1920 requiring more disjoint variables. (Contributed by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on . (Revised by Steven Nguyen, 13-Aug-2023.) |
Theorem | sbco2v 1922* | Version of sbco2 1939 with disjoint variable conditions. (Contributed by Wolf Lammen, 29-Apr-2023.) |
Theorem | hbsb 1923* | If is not free in , it is not free in when and are distinct. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.) |
Theorem | equsb3lem 1924* | Lemma for equsb3 1925. (Contributed by NM, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Theorem | equsb3 1925* | Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.) |
Theorem | sbn 1926 | Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.) |
Theorem | sbim 1927 | Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.) |
Theorem | sbor 1928 | Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.) |
Theorem | sban 1929 | Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.) |
Theorem | sbrim 1930 | Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.) |
Theorem | sblim 1931 | Substitution with a variable not free in consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | sb3an 1932 | Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.) |
Theorem | sbbi 1933 | Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.) |
Theorem | sblbis 1934 | Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.) |
Theorem | sbrbis 1935 | Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) |
Theorem | sbrbif 1936 | Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) |
Theorem | sbco2yz 1937* | This is a version of sbco2 1939 where is distinct from . It is a lemma on the way to proving sbco2 1939 which has no distinct variable constraints. (Contributed by Jim Kingdon, 19-Mar-2018.) |
Theorem | sbco2h 1938 | A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.) |
Theorem | sbco2 1939 | A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbco2d 1940 | A composition law for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbco2vd 1941* | Version of sbco2d 1940 with a distinct variable constraint between and . (Contributed by Jim Kingdon, 19-Feb-2018.) |
Theorem | sbco 1942 | A composition law for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbco3v 1943* | Version of sbco3 1948 with a distinct variable constraint between and . (Contributed by Jim Kingdon, 19-Feb-2018.) |
Theorem | sbcocom 1944 | Relationship between composition and commutativity for substitution. (Contributed by Jim Kingdon, 28-Feb-2018.) |
Theorem | sbcomv 1945* | Version of sbcom 1949 with a distinct variable constraint between and . (Contributed by Jim Kingdon, 28-Feb-2018.) |
Theorem | sbcomxyyz 1946* | Version of sbcom 1949 with distinct variable constraints between and , and and . (Contributed by Jim Kingdon, 21-Mar-2018.) |
Theorem | sbco3xzyz 1947* | Version of sbco3 1948 with distinct variable constraints between and , and and . Lemma for proving sbco3 1948. (Contributed by Jim Kingdon, 22-Mar-2018.) |
Theorem | sbco3 1948 | A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.) |
Theorem | sbcom 1949 | A commutativity law for substitution. (Contributed by NM, 27-May-1997.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.) |
Theorem | nfsbt 1950* | Closed form of nfsb 1920. (Contributed by Jim Kingdon, 9-May-2018.) |
Theorem | nfsbd 1951* | Deduction version of nfsb 1920. (Contributed by NM, 15-Feb-2013.) |
Theorem | elsb3 1952* | Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Theorem | elsb4 1953* | Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Theorem | sb9v 1954* | Like sb9 1955 but with a distinct variable constraint between and . (Contributed by Jim Kingdon, 28-Feb-2018.) |
Theorem | sb9 1955 | Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.) |
Theorem | sb9i 1956 | Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.) |
Theorem | sbnf2 1957* | 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 1958* | Deduction version of hbsb 1923. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.) |
Theorem | 2sb5 1959* | Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
Theorem | 2sb6 1960* | Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
Theorem | sbcom2v 1961* | Lemma for proving sbcom2 1963. It is the same as sbcom2 1963 but with additional distinct variable constraints on and , and on and . (Contributed by Jim Kingdon, 19-Feb-2018.) |
Theorem | sbcom2v2 1962* | Lemma for proving sbcom2 1963. It is the same as sbcom2v 1961 but removes the distinct variable constraint on and . (Contributed by Jim Kingdon, 19-Feb-2018.) |
Theorem | sbcom2 1963* | 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 1964* | Equivalence for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | 2sb5rf 1965* | Reversed double substitution. (Contributed by NM, 3-Feb-2005.) |
Theorem | 2sb6rf 1966* | Reversed double substitution. (Contributed by NM, 3-Feb-2005.) |
Theorem | dfsb7 1967* | An alternate definition of proper substitution df-sb 1737. 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 1860, first for then for . Compare Definition 2.1'' of [Quine] p. 17. Theorem sb7f 1968 provides a version where and don't have to be distinct. (Contributed by NM, 28-Jan-2004.) |
Theorem | sb7f 1968* | This version of dfsb7 1967 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 1507 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1737 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 1969* | An alternate definition of proper substitution df-sb 1737. Similar to dfsb7a 1970 but does not require that and be distinct. Similar to sb7f 1968 in that it involves a dummy variable , but expressed in terms of rather than . (Contributed by Jim Kingdon, 5-Feb-2018.) |
Theorem | dfsb7a 1970* | An alternate definition of proper substitution df-sb 1737. Similar to dfsb7 1967 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 1969. (Contributed by Jim Kingdon, 5-Feb-2018.) |
Theorem | sb10f 1971* | 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 1972* | 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 1973* | Elimination of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbel2x 1974* | Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbalyz 1975* | Move universal quantifier in and out of substitution. Identical to sbal 1976 except that it has an additional distinct variable constraint on and . (Contributed by Jim Kingdon, 29-Dec-2017.) |
Theorem | sbal 1976* | Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.) |
Theorem | sbal1yz 1977* | Lemma for proving sbal1 1978. Same as sbal1 1978 but with an additional disjoint variable condition on . (Contributed by Jim Kingdon, 23-Feb-2018.) |
Theorem | sbal1 1978* | A theorem used in elimination of disjoint variable conditions on by replacing it with a distinctor . (Contributed by NM, 5-Aug-1993.) (Proof rewitten by Jim Kingdon, 24-Feb-2018.) |
Theorem | sbexyz 1979* | Move existential quantifier in and out of substitution. Identical to sbex 1980 except that it has an additional disjoint variable condition on . (Contributed by Jim Kingdon, 29-Dec-2017.) |
Theorem | sbex 1980* | Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.) |
Theorem | sbalv 1981* | Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.) |
Theorem | sbco4lem 1982* | Lemma for sbco4 1983. It replaces the temporary variable with another temporary variable . (Contributed by Jim Kingdon, 26-Sep-2018.) |
Theorem | sbco4 1983* | 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 1984* | An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.) |
Theorem | 2exsb 1985* | An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) |
Theorem | dvelimALT 1986* | Version of dvelim 1993 that doesn't use ax-10 1484. Because it has different distinct variable constraints than dvelim 1993 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 1987* | Like dvelimf 1991 but with a distinct variable constraint on and . (Contributed by Jim Kingdon, 6-Mar-2018.) |
Theorem | hbsb4 1988 | 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 1989 | A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1988). (Contributed by NM, 7-Apr-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | nfsb4t 1990 | A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1988). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof rewritten by Jim Kingdon, 9-May-2018.) |
Theorem | dvelimf 1991 | Version of dvelim 1993 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) |
Theorem | dvelimdf 1992 | Deduction form of dvelimf 1991. This version may be useful if we want to avoid ax-17 1507 and use ax-16 1787 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) |
Theorem | dvelim 1993* |
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 1992. Other variants of this theorem are dvelimf 1991 (with no distinct variable restrictions) and dvelimALT 1986 (that avoids ax-10 1484). (Contributed by NM, 23-Nov-1994.) |
Theorem | dvelimor 1994* | 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 1995* | 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 1996* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) |
Theorem | dveel2 1997* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) |
Theorem | sbal2 1998* | Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) |
Theorem | nfsb4or 1999 | A variable not free remains so after substitution with a distinct variable. (Contributed by Jim Kingdon, 11-May-2018.) |
Syntax | weu 2000 | Extend wff definition to include existential uniqueness ("there exists a unique such that "). |
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