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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cbvexdva 1801* | 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 1802* | Rule used to change bound variables, using implicit substitition. (Contributed by NM, 26-Jul-1995.) |
Theorem | eean 1803 | Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | eeanv 1804* | Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) |
Theorem | eeeanv 1805* | Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | ee4anv 1806* | Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.) |
Theorem | ee8anv 1807* | Rearrange existential quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.) |
Theorem | nexdv 1808* | Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.) |
Theorem | chvarv 1809* | Implicit substitution of for into a theorem. (Contributed by NM, 20-Apr-1994.) |
Theorem | cleljust 1810* | 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 1391 with the class variables in wcel 1390. (Contributed by NM, 28-Jan-2004.) |
Theorem | hbs1 1811* | 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 1812* | is not free in when and are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | sbhb 1813* | Two ways of expressing " is (effectively) not free in ." (Contributed by NM, 29-May-2009.) |
Theorem | hbsbv 1814* | This is a version of hbsb 1820 with an extra distinct variable constraint, on and . (Contributed by Jim Kingdon, 25-Dec-2017.) |
Theorem | nfsbxy 1815* | Similar to hbsb 1820 but with an extra distinct variable constraint, on and . (Contributed by Jim Kingdon, 19-Mar-2018.) |
Theorem | nfsbxyt 1816* | Closed form of nfsbxy 1815. (Contributed by Jim Kingdon, 9-May-2018.) |
Theorem | sbco2vlem 1817* | This is a version of sbco2 1836 where is distinct from and from . It is a lemma on the way to proving sbco2v 1818 which only requires that and be distinct. (Contributed by Jim Kingdon, 25-Dec-2017.) (One distinct variable constraint removed by Jim Kingdon, 3-Feb-2018.) |
Theorem | sbco2v 1818* | This is a version of sbco2 1836 where is distinct from . (Contributed by Jim Kingdon, 12-Feb-2018.) |
Theorem | nfsb 1819* | 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 | hbsb 1820* | 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 1821* | Lemma for equsb3 1822. (Contributed by NM, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Theorem | equsb3 1822* | Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.) |
Theorem | sbn 1823 | Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.) |
Theorem | sbim 1824 | Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.) |
Theorem | sbor 1825 | 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 1826 | 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 1827 | Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.) |
Theorem | sblim 1828 | 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 1829 | Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.) |
Theorem | sbbi 1830 | Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.) |
Theorem | sblbis 1831 | Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.) |
Theorem | sbrbis 1832 | Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) |
Theorem | sbrbif 1833 | Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) |
Theorem | sbco2yz 1834* | This is a version of sbco2 1836 where is distinct from . It is a lemma on the way to proving sbco2 1836 which has no distinct variable constraints. (Contributed by Jim Kingdon, 19-Mar-2018.) |
Theorem | sbco2h 1835 | A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.) |
Theorem | sbco2 1836 | A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbco2d 1837 | A composition law for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbco2vd 1838* | Version of sbco2d 1837 with a distinct variable constraint between and . (Contributed by Jim Kingdon, 19-Feb-2018.) |
Theorem | sbco 1839 | A composition law for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbco3v 1840* | Version of sbco3 1845 with a distinct variable constraint between and . (Contributed by Jim Kingdon, 19-Feb-2018.) |
Theorem | sbcocom 1841 | Relationship between composition and commutativity for substitution. (Contributed by Jim Kingdon, 28-Feb-2018.) |
Theorem | sbcomv 1842* | Version of sbcom 1846 with a distinct variable constraint between and . (Contributed by Jim Kingdon, 28-Feb-2018.) |
Theorem | sbcomxyyz 1843* | Version of sbcom 1846 with distinct variable constraints between and , and and . (Contributed by Jim Kingdon, 21-Mar-2018.) |
Theorem | sbco3xzyz 1844* | Version of sbco3 1845 with distinct variable constraints between and , and and . Lemma for proving sbco3 1845. (Contributed by Jim Kingdon, 22-Mar-2018.) |
Theorem | sbco3 1845 | A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.) |
Theorem | sbcom 1846 | A commutativity law for substitution. (Contributed by NM, 27-May-1997.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.) |
Theorem | nfsbt 1847* | Closed form of nfsb 1819. (Contributed by Jim Kingdon, 9-May-2018.) |
Theorem | nfsbd 1848* | Deduction version of nfsb 1819. (Contributed by NM, 15-Feb-2013.) |
Theorem | elsb3 1849* | Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Theorem | elsb4 1850* | Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Theorem | sb9v 1851* | Like sb9 1852 but with a distinct variable constraint between and . (Contributed by Jim Kingdon, 28-Feb-2018.) |
Theorem | sb9 1852 | Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.) |
Theorem | sb9i 1853 | Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.) |
Theorem | sbnf2 1854* | 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 1855* | Deduction version of hbsb 1820. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.) |
Theorem | 2sb5 1856* | Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
Theorem | 2sb6 1857* | Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
Theorem | sbcom2v 1858* | Lemma for proving sbcom2 1860. It is the same as sbcom2 1860 but with additional distinct variable constraints on and , and on and . (Contributed by Jim Kingdon, 19-Feb-2018.) |
Theorem | sbcom2v2 1859* | Lemma for proving sbcom2 1860. It is the same as sbcom2v 1858 but removes the distinct variable constraint on and . (Contributed by Jim Kingdon, 19-Feb-2018.) |
Theorem | sbcom2 1860* | 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 1861* | Equivalence for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | 2sb5rf 1862* | Reversed double substitution. (Contributed by NM, 3-Feb-2005.) |
Theorem | 2sb6rf 1863* | Reversed double substitution. (Contributed by NM, 3-Feb-2005.) |
Theorem | dfsb7 1864* | An alternate definition of proper substitution df-sb 1643. 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 1764, first for then for . Compare Definition 2.1'' of [Quine] p. 17. Theorem sb7f 1865 provides a version where and don't have to be distinct. (Contributed by NM, 28-Jan-2004.) |
Theorem | sb7f 1865* | This version of dfsb7 1864 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 1416 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1643 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 1866* | An alternative definition of proper substitution df-sb 1643. Similar to dfsb7a 1867 but does not require that and be distinct. Similar to sb7f 1865 in that it involves a dummy variable , but expressed in terms of rather than . (Contributed by Jim Kingdon, 5-Feb-2018.) |
Theorem | dfsb7a 1867* | An alternative definition of proper substitution df-sb 1643. Similar to dfsb7 1864 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 1866. (Contributed by Jim Kingdon, 5-Feb-2018.) |
Theorem | sb10f 1868* | 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 1869* | 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 1870* | Elimination of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbel2x 1871* | Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbalyz 1872* | Move universal quantifier in and out of substitution. Identical to sbal 1873 except that it has an additional distinct variable constraint on and . (Contributed by Jim Kingdon, 29-Dec-2017.) |
Theorem | sbal 1873* | Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.) |
Theorem | sbal1yz 1874* | Lemma for proving sbal1 1875. Same as sbal1 1875 but with an additional distinct variable constraint on and . (Contributed by Jim Kingdon, 23-Feb-2018.) |
Theorem | sbal1 1875* | 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 1876* | Move existential quantifier in and out of substitution. Identical to sbex 1877 except that it has an additional distinct variable constraint on and . (Contributed by Jim Kingdon, 29-Dec-2017.) |
Theorem | sbex 1877* | Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.) |
Theorem | sbalv 1878* | Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.) |
Theorem | sbco4lem 1879* | Lemma for sbco4 1880. It replaces the temporary variable with another temporary variable . (Contributed by Jim Kingdon, 26-Sep-2018.) |
Theorem | sbco4 1880* | 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 1881* | An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.) |
Theorem | 2exsb 1882* | An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) |
Theorem | dvelimALT 1883* | Version of dvelim 1890 that doesn't use ax-10 1393. Because it has different distinct variable constraints than dvelim 1890 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 1884* | Like dvelimf 1888 but with a distinct variable constraint on and . (Contributed by Jim Kingdon, 6-Mar-2018.) |
Theorem | hbsb4 1885 | 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 1886 | A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1885). (Contributed by NM, 7-Apr-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | nfsb4t 1887 | A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1885). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof rewritten by Jim Kingdon, 9-May-2018.) |
Theorem | dvelimf 1888 | Version of dvelim 1890 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) |
Theorem | dvelimdf 1889 | Deduction form of dvelimf 1888. This version may be useful if we want to avoid ax-17 1416 and use ax-16 1692 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) |
Theorem | dvelim 1890* |
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 1889. Other variants of this theorem are dvelimf 1888 (with no distinct variable restrictions) and dvelimALT 1883 (that avoids ax-10 1393). (Contributed by NM, 23-Nov-1994.) |
Theorem | dvelimor 1891* | 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 1892* | 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 1893* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) |
Theorem | dveel2 1894* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) |
Theorem | sbal2 1895* | Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) |
Theorem | nfsb4or 1896 | A variable not free remains so after substitution with a distinct variable. (Contributed by Jim Kingdon, 11-May-2018.) |
Syntax | weu 1897 | Extend wff definition to include existential uniqueness ("there exists a unique such that "). |
Syntax | wmo 1898 | Extend wff definition to include uniqueness ("there exists at most one such that "). |
Theorem | eujust 1899* | A soundness justification theorem for df-eu 1900, 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 1900* | 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 1922, eu2 1941, eu3 1943, and eu5 1944 (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 1990). (Contributed by NM, 5-Aug-1993.) |
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