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Theorem List for Metamath Proof Explorer - 1801-1900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremexlimi 1801 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theorem19.23bi 1802 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theoremexlimd 1803 Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremexlimdh 1804 Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jan-1997.)

Theorem19.27 1805 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.28 1806 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.36 1807 Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.36i 1808 Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.37 1809 Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.38 1810 Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.32 1811 Theorem 19.32 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Theorem19.31 1812 Theorem 19.31 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.44 1813 Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.45 1814 Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.41 1815 Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem19.42 1816 Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)

Theoremexcom13 1817 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)

Theoremexrot3 1818 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)

Theoremexrot4 1819 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)

Theoremnexr 1820 Inference from 19.8a 1718. (Contributed by Jeff Hankins, 26-Jul-2009.)

Theoremnfim1 1821 A closed form of nfim 1769. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Theoremnfan1 1822 A closed form of nfan 1771. (Contributed by Mario Carneiro, 3-Oct-2016.)

Theoremexan 1823 Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremhbnd 1824 Deduction form of bound-variable hypothesis builder hbn 1720. (Contributed by NM, 3-Jan-2002.)

Theoremaaan 1825 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)

Theoremeeor 1826 Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)

Theoremqexmid 1827 Quantified "excluded middle." Exercise 9.2a of Boolos, p. 111, Computability and Logic. (Contributed by NM, 10-Dec-2000.)

Theoremequs5a 1828 A property related to substitution that unlike equs5 1936 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Theoremequs5e 1829 A property related to substitution that unlike equs5 1936 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Theoremexlimdd 1830 Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theorem19.21v 1831* 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 1791 via the use of distinct variable conditions combined with nfv 1605. 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 2149 derived from df-eu 2147. The "f" stands for "not free in" which is less restrictive than "does not occur in." (Contributed by NM, 5-Aug-1993.)

Theorem19.23v 1832* Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jun-1998.)

Theorem19.23vv 1833* Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.)

Theorempm11.53 1834* Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)

Theorem19.27v 1835* Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.)

Theorem19.28v 1836* Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.)

Theorem19.36v 1837* Special case of Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)

Theorem19.36aiv 1838* Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.12vv 1839* Special case of 19.12 1734 where its converse holds. (Contributed by NM, 18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.)

Theorem19.37v 1840* Special case of Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.37aiv 1841* Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.41v 1842* Special case of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.41vv 1843* Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.)

Theorem19.41vvv 1844* Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 30-Apr-1995.)

Theorem19.41vvvv 1845* Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers. (Contributed by FL, 14-Jul-2007.)

Theorem19.42v 1846* Special case of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theoremexdistr 1847* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)

Theorem19.42vv 1848* Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 16-Mar-1995.)

Theorem19.42vvv 1849* Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 21-Sep-2011.)

Theoremexdistr2 1850* Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)

Theorem3exdistr 1851* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem4exdistr 1852* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)

Theoremeean 1853 Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremeeanv 1854* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)

Theoremeeeanv 1855* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremee4anv 1856* Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.)

Theoremnexdv 1857* Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.)

Theoremstdpc7 1858 One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1650.) Translated to traditional notation, it can be read: " , provided that is free for in ." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)

Theoremsbequ1 1859 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ12 1860 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ12r 1861 An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremsbequ12a 1862 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbid 1863 An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.)

Theoremsb4a 1864 A version of sb4 1993 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Theoremsb4e 1865 One direction of a simplified definition of substitution that unlike sb4 1993 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

1.5.4  Axiom scheme ax-12 (Quantified Equality)

Axiomax-12 1866 Axiom of Quantified Equality. One of the equality and substitution axioms of predicate calculus with equality.

An equivalent way to express this axiom that may be easier to understand is (see ax12b 1655). Recall that in the intended interpretation, our variables are metavariables ranging over the variables of predicate calculus (the object language). In order for the first antecedent to hold, and must have different values and thus cannot be the same object-language variable. Similarly, and cannot be the same object-language variable. Therefore, will not occur in the wff when the first two antecedents hold, so analogous to ax-17 1603, the conclusion follows.

The original version of this axiom was ax-12o 2081 and was replaced with this shorter ax-12 1866 in December 2015. The old axiom is proved from this one as theorem ax12o 1875. Conversely, this axiom is proved from ax-12o 2081 as theorem ax12 2095.

The primary purpose of this axiom is to provide a way to introduce the quantifier on even when and are substituted with the same variable. In this case, the first antecedent becomes and the axiom still holds.

Although this version is shorter, the original version ax12o 1875 may be more practical to work with because of the "distinctor" form of its antecedents. A typical application of ax12o 1875 is in dvelimh 1904 which converts a distinct variable pair to the distinctor antecendent .

This axiom can be weakened if desired by adding distinct variable restrictions on pairs and . To show that, we add these restrictions to theorem ax12v 1867 and use only ax12v 1867 for further derivations. Thus ax12v 1867 should be the only theorem referencing this axiom. Other theorems can reference either ax12v 1867 or ax12o 1875.

This axiom scheme is logically redundant (see ax12w 1698) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.)

Theoremax12v 1867* A weaker version of ax-12 1866 with distinct variable restrictions on pairs and . In order to show that this weakening is adequate, this should be the only theorem referencing ax-12 1866 directly. (Contributed by NM, 30-Jun-2016.)

Theoremax12olem1 1868* Lemma for ax12o 1875. Similar to equvin 1941 but with a negated equality. (Contributed by NM, 24-Dec-2015.)

Theoremax12olem2 1869* Lemma for ax12o 1875. Negate the equalities in ax-12 1866, shown as the hypothesis. (Contributed by NM, 24-Dec-2015.)

Theoremax12olem3 1870 Lemma for ax12o 1875. Show the equivalence of an intermediate equivalent to ax12o 1875 with the conjunction of ax-12 1866 and a variant with negated equalities. (Contributed by NM, 24-Dec-2015.)

Theoremax12olem4 1871* Lemma for ax12o 1875. Construct an intermediate equivalent to ax-12 1866 from two instances of ax-12 1866. (Contributed by NM, 24-Dec-2015.)

Theoremax12olem5 1872 Lemma for ax12o 1875. See ax12olem6 1873 for derivation of ax12o 1875 from the conclusion. (Contributed by NM, 24-Dec-2015.)

Theoremax12olem6 1873* Lemma for ax12o 1875. Derivation of ax12o 1875 from the hypotheses, without using ax12o 1875. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 24-Dec-2015.)

Theoremax12olem7 1874* Lemma for ax12o 1875. Derivation of ax12o 1875 from the hypotheses, without using ax12o 1875. (Contributed by NM, 24-Dec-2015.)

Theoremax12o 1875 Derive set.mm's original ax-12o 2081 from the shorter ax-12 1866. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.)

Theoremax10lem1 1876* Lemma for ax10 1884. Change bound variable. (Contributed by NM, 22-Jul-2015.)

Theoremax10lem2 1877* Lemma for ax10 1884. Change free variable. (Contributed by NM, 25-Jul-2015.)

Theoremax10lem3 1878* Lemma for ax10 1884. Similar to ax-10 2079 but with distinct variables. (Contributed by NM, 25-Jul-2015.)

Theoremdvelimv 1879* Similar to dvelim 1956 with first hypothesis replaced by distinct variable condition. (Contributed by NM, 25-Jul-2015.)

Theoremdveeq2 1880* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.)

Theoremax10lem4 1881* Lemma for ax10 1884. Change bound variable. (Contributed by NM, 8-Jul-2016.)

Theoremax10lem5 1882* Lemma for ax10 1884. Change free and bound variables. (Contributed by NM, 22-Jul-2015.)

Theoremax10lem6 1883 Lemma for ax10 1884. Similar to ax10o 1892 but with reversed antecedent. (Contributed by NM, 25-Jul-2015.)

Theoremax10 1884 Derive set.mm's original ax-10 2079 from others. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 7-Nov-2015.)

Theorema16g 1885* Generalization of ax16 1985. (Contributed by NM, 25-Jul-2015.)

Theoremaecom 1886 Commutation law for identical variable specifiers. The antecedent and consequent are true when and are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.)

Theoremaecoms 1887 A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)

Theoremnaecoms 1888 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)

Theoremax9 1889 Theorem showing that ax-9 1635 follows from the weaker version ax9v 1636. (Even though this theorem depends on ax-9 1635, all references of ax-9 1635 are made via ax9v 1636. An earlier version stated ax9v 1636 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-9 1635 so that all proofs can be traced back to ax9v 1636. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.)

Theoremax9o 1890 Show that the original axiom ax-9o 2077 can be derived from ax9 1889 and others. See ax9from9o 2087 for the rederivation of ax9 1889 from ax-9o 2077.

Normally, ax9o 1890 should be used rather than ax-9o 2077, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.)

Theorema9e 1891 At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-5 1544 through ax-14 1688 and ax-17 1603, all axioms other than ax9 1889 are believed to be theorems of free logic, although the system without ax9 1889 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.)

Theoremax10o 1892 Show that ax-10o 2078 can be derived from ax-10 2079 in the form of ax10 1884. Normally, ax10o 1892 should be used rather than ax-10o 2078, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.)

Theoremhbae 1893 All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.)

Theoremnfae 1894 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbnae 1895 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.)

Theoremnfnae 1896 All variables are effectively bound in an distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbnaes 1897 Rule that applies hbnae 1895 to antecedent. (Contributed by NM, 5-Aug-1993.)

Theoremnfeqf 1898 A variable is effectively not free in an equality if it is not either of the involved variables. version of ax-12o 2081. (Contributed by Mario Carneiro, 6-Oct-2016.)

Theoremequs4 1899 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)

Theoremequsal 1900 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.)

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