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

Theoremspeimfw 1801 Specialization, with additional weakening (compared to 19.2 1817) to allow bundling of and . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Dec-2017.)

TheoremspeimfwALT 1802 Alternate proof of speimfw 1801 (longer compressed proof, but fewer essential steps). (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Aug-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremspimfw 1803 Specialization, with additional weakening (compared to sp 1957) to allow bundling of and . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)

Theoremax12i 1804 Inference that has ax-12 1950 (without ) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without using ax-12 1950 in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)

1.4.6  Define proper substitution

Syntaxwsb 1805 Extend wff definition to include proper substitution (read "the wff that results when is properly substituted for in wff "). (Contributed by NM, 24-Jan-2006.)

Definitiondf-sb 1806 Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use to mean "the wff that results from the proper substitution of for in the wff ." That is, properly replaces . For example, is the same as , as shown in elsb4 2284. We can also use in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2202.

Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, " is the wff that results when is properly substituted for in ." For example, if the original is , then is , from which we obtain that is . So what exactly does mean? Curry's notation solves this problem.

In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 2225, sbcom2 2294 and sbid2v 2306).

Note that our definition is valid even when and are replaced with the same variable, as sbid 2101 shows. We achieve this by having free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 2304 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another version that mixes free and bound variables is dfsb3 2223. When and are distinct, we can express proper substitution with the simpler expressions of sb5 2279 and sb6 2278.

There are no restrictions on any of the variables, including what variables may occur in wff . (Contributed by NM, 10-May-1993.)

Theoremsbequ2 1807 An equality theorem for substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.)

Theoremsb1 1808 One direction of a simplified definition of substitution. The converse requires either a dv condition (sb5 2279) or a non-freeness hypothesis (sb5f 2235). (Contributed by NM, 13-May-1993.)

Theoremspsbe 1809 A specialization theorem. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.)

Theoremsbequ8 1810 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 28-Jul-2018.)

Theoremsbimi 1811 Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)

Theoremsbbii 1812 Infer substitution into both sides of a logical equivalence. (Contributed by NM, 14-May-1993.)

1.4.7  Axiom scheme ax-6 (Existence)

Axiomax-6 1813 Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. This axiom tells us is that at least one thing exists. In this form (not requiring that and be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) It is equivalent to axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint); the equivalence is established by axc10 2109 and ax6fromc10 32532. A more convenient form of this axiom is ax6e 2107, which has additional remarks.

Raph Levien proved the independence of this axiom from the other logical axioms on 12-Apr-2005. See item 16 at http://us.metamath.org/award2003.html.

ax-6 1813 can be proved from the weaker version ax6v 1814 requiring that the variables be distinct; see theorem ax6 2108.

ax-6 1813 can also be proved from the Axiom of Separation (in the form that we use that axiom, where free variables are not universally quantified). See theorem ax6vsep 4522.

Except by ax6v 1814, this axiom should not be referenced directly. Instead, use theorem ax6 2108. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

Theoremax6v 1814* Axiom B7 of [Tarski] p. 75, which requires that and be distinct. This trivial proof is intended merely to weaken axiom ax-6 1813 by adding a distinct variable restriction. From here on, ax-6 1813 should not be referenced directly by any other proof, so that theorem ax6 2108 will show that we can recover ax-6 1813 from this weaker version if it were an axiom (as it is in the case of Tarski).

Note: Introducing as a distinct variable group "out of the blue" with no apparent justification has puzzled some people, but it is perfectly sound. All we are doing is adding an additional redundant requirement, no different from adding a redundant logical hypothesis, that results in a weakening of the theorem. This means that any future theorem that references ax6v 1814 must have a \$d specified for the two variables that get substituted for and . The \$d does not propagate "backwards" i.e. it does not impose a requirement on ax-6 1813.

When possible, use of this theorem rather than ax6 2108 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 7-Aug-2015.)

Theoremax6ev 1815* At least one individual exists. Weaker version of ax6e 2107. When possible, use of this theorem rather than ax6e 2107 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 3-Aug-2017.)

Theoremexiftru 1816 Rule of existential generalization, similar to universal generalization ax-gen 1677, but valid only if an individual exists. Its proof requires ax-6 1813 but the equality predicate does not occur in its statement. Some fundamental theorems of predicate logic can be proven from ax-gen 1677, ax-4 1690 and this theorem alone, not requiring ax-7 1859 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 9-Dec-2017.)

Theorem19.2 1817 Theorem 19.2 of [Margaris] p. 89. This corresponds to the axiom (D) of modal logic. Note: This proof is very different from Margaris' because we only have Tarski's FOL axiom schemes available at this point. See the later 19.2g 1966 for a more conventional proof of a more general result, which uses additional axioms. (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 1859. (Revised by Wolf Lammen, 4-Dec-2017.)

Theorem19.8w 1818 Weak version of 19.8a 1955. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.)

Theorem19.8v 1819* Version of 19.8a 1955 with a dv condition, requiring fewer axioms. (Contributed by BJ, 12-Mar-2020.)

Theorem19.9v 1820* Version of 19.9 1990 with a dv condition, requiring fewer axioms. Any formula can be existentially quantified using a variable which it does not contain. See also 19.3v 1821. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 1859. (Revised by Wolf Lammen, 4-Dec-2017.)

Theorem19.3v 1821* Version of 19.3 1986 with a dv condition, requiring fewer axioms. Any formula can be universally quantified using a variable which it does not contain. See also 19.9v 1820. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 1859. (Revised by Wolf Lammen, 4-Dec-2017.)

Theoremspvw 1822* Version of sp 1957 when does not occur in . Converse of ax-5 1766. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.)

Theorem19.39 1823 Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Theorem19.24 1824 Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Theorem19.34 1825 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Theorem19.23v 1826* Version of 19.23 2013 with a dv condition instead of a non-freeness hypothesis. (Contributed by NM, 28-Jun-1998.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 11-Jan-2020.)

Theorem19.23vv 1827* Theorem 19.23v 1826 extended to two variables. (Contributed by NM, 10-Aug-2004.)

Theorem19.36v 1828* Version of 19.36 2063 with a dv condition instead of a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.)

Theorem19.36iv 1829* Inference associated with 19.36v 1828. Version of 19.36i 2064 with a dv condition. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.)

Theorempm11.53v 1830* Version of pm11.53 2090 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)

Theorem19.12vvv 1831* Version of 19.12vv 2091 with a dv condition, requiring fewer axioms. See also 19.12 2052. (Contributed by BJ, 18-Mar-2020.)

Theorem19.27v 1832* Version of 19.27 2026 with a dv condition, requiring fewer axioms. (Contributed by NM, 3-Jun-2004.)

Theorem19.28v 1833* Version of 19.28 2027 with a dv condition, requiring fewer axioms. (Contributed by NM, 25-Mar-2004.)

Theorem19.37v 1834* Version of 19.37 2065 with a dv condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)

Theorem19.37iv 1835* Inference associated with 19.37v 1834. (Contributed by NM, 5-Aug-1993.)

Theorem19.44v 1836* Version of 19.44 2068 with a dv condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)

Theorem19.45v 1837* Version of 19.45 2069 with a dv condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)

Theorem19.41v 1838* Version of 19.41 2070 with a dv condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)

Theorem19.41vv 1839* Version of 19.41 2070 with two quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)

Theorem19.41vvv 1840* Version of 19.41 2070 with three quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)

Theorem19.41vvvv 1841* Version of 19.41 2070 with four quantifiers and a dv condition requiring fewer axioms. (Contributed by FL, 14-Jul-2007.)

Theorem19.42v 1842* Version of 19.42 2071 with a dv condition requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)

Theoremexdistr 1843* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)

Theorem19.42vv 1844* Version of 19.42 2071 with two quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 16-Mar-1995.)

Theorem19.42vvv 1845* Version of 19.42 2071 with three quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 21-Sep-2011.)

Theoremexdistr2 1846* Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)

Theorem3exdistr 1847* Distribution of existential quantifiers in a triple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem4exdistr 1848* Distribution of existential quantifiers in a quadruple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Wolf Lammen, 20-Jan-2018.)

Theoremspimeh 1849* Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)

Theoremspimw 1850* Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)

Theoremspimvw 1851* Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Theoremspnfw 1852 Weak version of sp 1957. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 13-Aug-2017.)

Theoremspfalw 1853 Version of sp 1957 when is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 25-Dec-2017.)

Theoremequs4v 1854* Version of equs4 2140 with a dv condition, which requires fewer axioms. (Contributed by BJ, 31-May-2019.)

Theoremequsalvw 1855* Version of equsal 2141 with two dv conditions, which requires fewer axioms. (Contributed by BJ, 31-May-2019.)

Theoremequsexvw 1856* Version of equsexv 2085 with a dv condition, which requires fewer axioms. See also equsex 2143. (Contributed by BJ, 31-May-2019.)

Theoremcbvaliw 1857* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.)

Theoremcbvalivw 1858* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)

1.4.8  Axiom scheme ax-7 (Equality)

Axiomax-7 1859 Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. It states that equality is right-Euclidean (this is similar, but not identical, to being transitive, which is proved as equtr 1873). This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom C7 of [Monk2] p. 105 and Axiom Scheme C8' in [Megill] p. 448 (p. 16 of the preprint).

The equality symbol was invented in 1557 by Robert Recorde. He chose a pair of parallel lines of the same length because "noe .2. thynges, can be moare equalle."

We prove in ax7 1868 that this axiom can be recovered from its weakened version ax7v 1860 where and are assumed to be disjoint variables. In particular, the only theorem referencing ax-7 1859 should be ax7v 1860. See the comment of ax7v 1860 for more details on these matters. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 7-Dec-2020.) Use ax7 1868 instead. (New usage is discouraged.)

Theoremax7v 1860* Weakened version of ax-7 1859, with a dv condition on . This should be the only proof referencing ax-7 1859, and it should be referenced only by its two weakened versions ax7v1 1861 and ax7v2 1862, from which ax-7 1859 is then rederived as ax7 1868, which shows that either ax7v 1860 or the conjunction of ax7v1 1861 and ax7v2 1862 is sufficient.

In ax7v 1860, it is still allowed to substitute the same variable for and , or the same variable for and . Therefore, ax7v 1860 "bundles" (a term coined by Raph Levien) its "principal instance" with distinct, and its "degenerate instances" and with distinct. These degenerate instances are for instance used in the proofs of equcomiv 1866 and equid 1863 respectively. (Contributed by BJ, 7-Dec-2020.) Use ax7 1868 instead. (New usage is discouraged.)

Theoremax7v1 1861* First of two weakened versions of ax7v 1860, with an extra dv condition on , see comments there. (Contributed by BJ, 7-Dec-2020.)

Theoremax7v2 1862* Second of two weakened versions of ax7v 1860, with an extra dv condition on , see comments there. (Contributed by BJ, 7-Dec-2020.)

Theoremequid 1863 Identity law for equality. Lemma 2 of [KalishMontague] p. 85. See also Lemma 6 of [Tarski] p. 68. (Contributed by NM, 1-Apr-2005.) (Revised by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Feb-2018.) (Proof shortened by Wolf Lammen, 22-Aug-2020.)

TheoremequidOLD 1864 Obsolete proof of equid 1863 as of 22-Aug-2020. (Contributed by NM, 1-Apr-2005.) (Revised by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Feb-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnfequid 1865 Bound-variable hypothesis builder for . This theorem tells us that any variable, including , is effectively not free in , even though is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)

Theoremequcomiv 1866* Weaker form of equcomi 1869 with a dv condition on . This is an intermediate step and equcomi 1869 is fully recovered later. (Contributed by BJ, 7-Dec-2020.)

Theoremax6evr 1867* A commuted form of ax6ev 1815. (Contributed by BJ, 7-Dec-2020.)

Theoremax7 1868 Proof of ax-7 1859 from ax7v1 1861 and ax7v2 1862, proving sufficiency of the conjunction of the latter two weakened versions of ax7v 1860, which is itself a weakened version of ax-7 1859.

Note that the weakened version of ax-7 1859 obtained by adding a dv condition on (resp. on ) does not permit, together with the other axioms, to prove reflexivity (resp. symmetry). (Contributed by BJ, 7-Dec-2020.)

Theoremequcomi 1869 Commutative law for equality. Equality is a symmetric relation. Lemma 3 of [KalishMontague] p. 85. See also Lemma 7 of [Tarski] p. 69. (Contributed by NM, 10-Jan-1993.) (Revised by NM, 9-Apr-2017.)

Theoremequcom 1870 Commutative law for equality. Equality is a symmetric relation. (Contributed by NM, 20-Aug-1993.)

Theoremequcomd 1871 Deduction form of equcom 1870, symmetry of equality. For the versions for classes, see eqcom 2478 and eqcomd 2477. (Contributed by BJ, 6-Oct-2019.)

Theoremequcoms 1872 An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 10-Jan-1993.)

Theoremequtr 1873 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)

Theoremequtrr 1874 A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)

Theoremequequ1 1875 An equivalence law for equality. (Contributed by NM, 1-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)

Theoremequequ2 1876 An equivalence law for equality. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.)

Theoremequtr2 1877 Equality is left-Euclidean. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremstdpc6 1878 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1879.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)

Theoremstdpc7 1879 One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1878.) 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.)

Theoremequviniv 1880* A specialized version of equvini 2195 with a distinct variable restriction. (Contributed by Wolf Lammen, 8-Sep-2018.)

Theoremequvin 1881* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 1932, ax-13 2104. (Revised by Wolf Lammen, 10-Jun-2019.)

Theoremax13b 1882 An equivalence used to show two ways of expressing ax-13 2104. See the comment for ax-13 2104. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.) (Revised by BJ, 15-Sep-2020.)

Theoremspfw 1883* Weak version of sp 1957. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. (Contributed by NM, 19-Apr-2017.)

Theoremspw 1884* Weak version of the specialization scheme sp 1957. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1957 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1957 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 1926 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1957 are spfw 1883 (minimal distinct variable requirements), spnfw 1852 (when is not free in ), spvw 1822 (when does not appear in ), sptruw 1689 (when is true), and spfalw 1853 (when is false). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)

Theoremcbvalw 1885* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Theoremcbvalvw 1886* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)

Theoremcbvexvw 1887* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)

Theoremalcomiw 1888* Weak version of alcom 1940. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.)

Theoremhbn1fw 1889* Weak version of ax-10 1932 from which we can prove any ax-10 1932 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)

Theoremhbn1w 1890* Weak version of hbn1 1933. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Theoremhba1w 1891* Weak version of hba1 1998. See comments for ax10w 1920. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Theoremhbe1w 1892* Weak version of hbe1 1934. See comments for ax10w 1920. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)

Theoremhbalw 1893* Weak version of hbal 1939. Uses only Tarski's FOL axiom schemes. Unlike hbal 1939, this theorem requires that and be distinct i.e. are not bundled. (Contributed by NM, 19-Apr-2017.)

Theoremspaev 1894* A special instance of sp 1957 applied to an equality with a dv condition. Unlike the more general sp 1957, we can prove this without ax-12 1950. This is an instance of aeveq 1898.

The antecedent with distinct and is a characteristic of a degenerate universe, in which just one object exists. Actually more than one object may still exist, but if so, we give up on equality as a discriminating term.

Separating this degenerate case from a richer universe, where inequality is possible, is a common proof idea. The name of this theorem follows a convention, where the condition is denoted by 'aev', a shorthand for 'all equal, with a distinct variable condition'. (Contributed by Wolf Lammen, 14-Mar-2021.)

Theoremcbvaev 1895* Change bound variable in an equality with a dv condition. This is an instance of aev 1899. (Contributed by NM, 22-Jul-2015.) (Revised by BJ, 18-Jun-2019.)

Theoremaevlem0 1896* Lemma for aevlem 1897. This is an instance of aev 1899. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-12 1950. (Revised by Wolf Lammen, 14-Mar-2021.) (Revised by BJ, 29-Mar-2021.)

Theoremaevlem 1897* Lemma for aev 1899 and axc16g 2042. Change free and bound variables. This is an instance of aev 1899. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2104, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 29-Mar-2021.)

Theoremaeveq 1898* with a dv restriction (One-object-universe) disallows inequality. (Contributed by Wolf Lammen, 19-Mar-2021.)

Theoremaev 1899* A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 1937. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2104, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 1950. (Revised by Wolf Lammen, 19-Mar-2021.)

Theoremhbaevg 1900* Generalization of hbaev 1901, proved at no extra cost. (Contributed by Wolf Lammen, 22-Mar-2021.) (Revised by BJ, 29-Mar-2021.)

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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