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Theorem List for Intuitionistic Logic Explorer - 1801-1900   *Has distinct variable group(s)
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Statement

1.4.3  More theorems related to ax-11 and substitution

Theoremalbidv 1801* Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)

Theoremexbidv 1802* Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)

Theoremax11b 1803 A bidirectional version of ax-11o 1800. (Contributed by NM, 30-Jun-2006.)

Theoremax11v 1804* This is a version of ax-11o 1800 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.)

Theoremax11ev 1805* Analogue to ax11v 1804 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)

Theoremequs5 1806 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)

Theoremequs5or 1807 Lemma used in proofs of substitution properties. Like equs5 1806 but, in intuitionistic logic, replacing negation and implication with disjunction makes this a stronger result. (Contributed by Jim Kingdon, 2-Feb-2018.)

Theoremsb3 1808 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)

Theoremsb4 1809 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)

Theoremsb4or 1810 One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1809 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.)

Theoremsb4b 1811 Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)

Theoremsb4bor 1812 Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.)

Theoremhbsb2 1813 Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremnfsb2or 1814 Bound-variable hypothesis builder for substitution. Similar to hbsb2 1813 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.)

Theoremsbequilem 1815 Propositional logic lemma used in the sbequi 1816 proof. (Contributed by Jim Kingdon, 1-Feb-2018.)

Theoremsbequi 1816 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.)

Theoremsbequ 1817 An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)

Theoremdrsb2 1818 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)

Theoremspsbe 1819 A specialization theorem, mostly the same as Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 29-Dec-2017.)

Theoremspsbim 1820 Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)

Theoremspsbbi 1821 Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)

Theoremsbbidh 1822 Deduction substituting both sides of a biconditional. New proofs should use sbbid 1823 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremsbbid 1823 Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.)

Theoremsbequ8 1824 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) (Proof revised by Jim Kingdon, 20-Jan-2018.)

Theoremsbft 1825 Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.)

Theoremsbid2h 1826 An identity law for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbid2 1827 An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsbidm 1828 An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)

Theoremsb5rf 1829 Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremsb6rf 1830 Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremsb8h 1831 Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)

Theoremsb8eh 1832 Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.)

Theoremsb8 1833 Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)

Theoremsb8e 1834 Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)

1.4.4  Predicate calculus with distinct variables (cont.)

Theoremax16i 1835* Inference with ax-16 1791 as its conclusion, that does not require ax-10 1482, ax-11 1483, or ax12 1489 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. (Contributed by NM, 20-May-2008.)

Theoremax16ALT 1836* Version of ax16 1790 that does not require ax-10 1482 or ax12 1489 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremspv 1837* Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.)

Theoremspimev 1838* Distinct-variable version of spime 1718. (Contributed by NM, 5-Aug-1993.)

Theoremspeiv 1839* Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.)

Theoremequvin 1840* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)

Theorema16g 1841* A generalization of Axiom ax-16 1791. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorema16gb 1842* A generalization of Axiom ax-16 1791. (Contributed by NM, 5-Aug-1993.)

Theorema16nf 1843* If there is only one element in the universe, then everything satisfies . (Contributed by Mario Carneiro, 7-Oct-2016.)

Theorem2albidv 1844* Formula-building rule for 2 existential quantifiers (deduction form). (Contributed by NM, 4-Mar-1997.)

Theorem2exbidv 1845* Formula-building rule for 2 existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.)

Theorem3exbidv 1846* Formula-building rule for 3 existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.)

Theorem4exbidv 1847* Formula-building rule for 4 existential quantifiers (deduction form). (Contributed by NM, 3-Aug-1995.)

Theorem19.9v 1848* Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.) (Revised by NM, 21-May-2007.)

Theoremexlimdd 1849 Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theorem19.21v 1850* Special case of Theorem 19.21 of [Margaris] p. 90. Notational convention: We sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as in 19.21 1560 via the use of distinct variable conditions combined with ax-17 1503. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the distinct variable condition; e.g., euf 2008 derived from df-eu 2006. The "f" stands for "not free in" which is less restrictive than "does not occur in". (Contributed by NM, 5-Aug-1993.)

Theoremalrimiv 1851* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theoremalrimivv 1852* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)

Theoremalrimdv 1853* Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.)

Theoremnfdv 1854* Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theorem2ax17 1855* Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)

Theoremalimdv 1856* Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.)

Theoremeximdv 1857* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)

Theorem2alimdv 1858* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.)

Theorem2eximdv 1859* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.)

Theorem19.23v 1860* Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jun-1998.)

Theorem19.23vv 1861* Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.)

Theoremsb56 1862* Two equivalent ways of expressing the proper substitution of for in , when and are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1740. (Contributed by NM, 14-Apr-2008.)

Theoremsb6 1863* Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)

Theoremsb5 1864* Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)

Theoremsbnv 1865* Version of sbn 1929 where and are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.)

Theoremsbanv 1866* Version of sban 1932 where and are distinct. (Contributed by Jim Kingdon, 24-Dec-2017.)

Theoremsborv 1867* Version of sbor 1931 where and are distinct. (Contributed by Jim Kingdon, 3-Feb-2018.)

Theoremsbi1v 1868* Forward direction of sbimv 1870. (Contributed by Jim Kingdon, 25-Dec-2017.)

Theoremsbi2v 1869* Reverse direction of sbimv 1870. (Contributed by Jim Kingdon, 18-Jan-2018.)

Theoremsbimv 1870* Intuitionistic proof of sbim 1930 where and are distinct. (Contributed by Jim Kingdon, 18-Jan-2018.)

Theoremsblimv 1871* Version of sblim 1934 where and are distinct. (Contributed by Jim Kingdon, 19-Jan-2018.)

Theorempm11.53 1872* Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)

Theoremexlimivv 1873* Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.)

Theoremexlimdvv 1874* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)

Theoremexlimddv 1875* Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)

Theorem19.27v 1876* Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.)

Theorem19.28v 1877* Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.)

Theorem19.36aiv 1878* Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.41v 1879* Special case of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.41vv 1880* Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.)

Theorem19.41vvv 1881* Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 30-Apr-1995.)

Theorem19.41vvvv 1882* Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers. (Contributed by FL, 14-Jul-2007.)

Theorem19.42v 1883* Special case of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theoremspvv 1884* Version of spv 1837 with a disjoint variable condition. (Contributed by BJ, 31-May-2019.)

Theoremchvarvv 1885* Version of chvarv 1914 with a disjoint variable condition. (Contributed by BJ, 31-May-2019.)

Theoremexdistr 1886* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)

Theoremexdistrv 1887* Distribute a pair of existential quantifiers (over disjoint variables) over a conjunction. Combination of 19.41v 1879 and 19.42v 1883. For a version with fewer disjoint variable conditions but requiring more axioms, see eeanv 1909. (Contributed by BJ, 30-Sep-2022.)

Theorem19.42vv 1888* Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 16-Mar-1995.)

Theorem19.42vvv 1889* Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 21-Sep-2011.)

Theorem19.42vvvv 1890* Theorem 19.42 of [Margaris] p. 90 with 4 quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)

Theoremexdistr2 1891* Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)

Theorem3exdistr 1892* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem4exdistr 1893* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)

Theoremcbvalv 1894* Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)

Theoremcbvexv 1895* Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)

Theoremcbvalvw 1896* Change bound variable. See cbvalv 1894 for a version with fewer disjoint variable conditions. (Contributed by NM, 9-Apr-2017.) Avoid ax-7 1425. (Revised by Gino Giotto, 25-Aug-2024.)

Theoremcbvexvw 1897* Change bound variable. See cbvexv 1895 for a version with fewer disjoint variable conditions. (Contributed by NM, 19-Apr-2017.) Avoid ax-7 1425. (Revised by Gino Giotto, 25-Aug-2024.)

Theoremcbval2 1898* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Apr-2018.)

Theoremcbvex2 1899* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremcbval2v 1900* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.)

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