Home Intuitionistic Logic ExplorerTheorem List (p. 22 of 106) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 2101-2200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsyl5req 2101 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Theoremsyl5eqr 2102 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5reqr 2103 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Theoremsyl6eq 2104 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)

Theoremsyl6req 2105 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Theoremsyl6eqr 2106 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)

Theoremsyl6reqr 2107 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Theoremsylan9eq 2108 An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremsylan9req 2109 An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)

Theoremsylan9eqr 2110 An equality transitivity deduction. (Contributed by NM, 8-May-1994.)

Theorem3eqtr3g 2111 A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)

Theorem3eqtr3a 2112 A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)

Theorem3eqtr4g 2113 A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.)

Theorem3eqtr4a 2114 A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremeq2tri 2115 A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)

Theoremeleq1 2116 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq2 2117 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq12 2118 Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)

Theoremeleq1i 2119 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq2i 2120 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq12i 2121 Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)

Theoremeleq1d 2122 Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq2d 2123 Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.)

Theoremeleq12d 2124 Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)

Theoremeleq1a 2125 A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)

Theoremeqeltri 2126 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Theoremeqeltrri 2127 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Theoremeleqtri 2128 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Theoremeleqtrri 2129 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Theoremeqeltrd 2130 Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremeqeltrrd 2131 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)

Theoremeleqtrd 2132 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)

Theoremeleqtrrd 2133 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)

Theorem3eltr3i 2134 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr4i 2135 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr3d 2136 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr4d 2137 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr3g 2138 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr4g 2139 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theoremsyl5eqel 2140 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl5eqelr 2141 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl5eleq 2142 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl5eleqr 2143 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eqel 2144 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eqelr 2145 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eleq 2146 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eleqr 2147 A membership and equality inference. (Contributed by NM, 24-Apr-2005.)

Theoremeleq2s 2148 Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremeqneltrd 2149 If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremeqneltrrd 2150 If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremneleqtrd 2151 If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremneleqtrrd 2152 If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremcleqh 2153* Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2217. (Contributed by NM, 5-Aug-1993.)

Theoremnelneq 2154 A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)

Theoremnelneq2 2155 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)

Theoremeqsb3lem 2156* Lemma for eqsb3 2157. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremeqsb3 2157* Substitution applied to an atomic wff (class version of equsb3 1841). (Contributed by Rodolfo Medina, 28-Apr-2010.)

Theoremclelsb3 2158* Substitution applied to an atomic wff (class version of elsb3 1868). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremclelsb4 2159* Substitution applied to an atomic wff (class version of elsb4 1869). (Contributed by Jim Kingdon, 22-Nov-2018.)

Theoremhbxfreq 2160 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1377 for equivalence version. (Contributed by NM, 21-Aug-2007.)

Theoremhblem 2161* Change the free variable of a hypothesis builder. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)

Theoremabeq2 2162* Equality of a class variable and a class abstraction (also called a class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the relationship between expressions with class abstractions and expressions with class variables. Note that abbi 2167 and its relatives are among those useful for converting theorems with class variables to equivalent theorems with wff variables, by first substituting a class abstraction for each class variable.

Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable (that has a free variable ) to a theorem with a class variable , we substitute for throughout and simplify, where is a new class variable not already in the wff. Conversely, to convert a theorem with a class variable to one with , we substitute for throughout and simplify, where and are new set and wff variables not already in the wff. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. (Contributed by NM, 5-Aug-1993.)

Theoremabeq1 2163* Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)

Theoremabeq2i 2164 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.)

Theoremabeq1i 2165 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)

Theoremabeq2d 2166 Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)

Theoremabbi 2167 Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)

Theoremabbi2i 2168* Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 5-Aug-1993.)

Theoremabbii 2169 Equivalent wff's yield equal class abstractions (inference rule). (Contributed by NM, 5-Aug-1993.)

Theoremabbid 2170 Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremabbidv 2171* Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 10-Aug-1993.)

Theoremabbi2dv 2172* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)

Theoremabbi1dv 2173* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)

Theoremabid2 2174* A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)

Theoremsb8ab 2175 Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)

Theoremcbvab 2176 Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremcbvabv 2177* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)

Theoremclelab 2178* Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)

Theoremclabel 2179* Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)

Theoremsbab 2180* The right-hand side of the second equality is a way of representing proper substitution of for into a class variable. (Contributed by NM, 14-Sep-2003.)

2.1.3  Class form not-free predicate

Syntaxwnfc 2181 Extend wff definition to include the not-free predicate for classes.

Theoremnfcjust 2182* Justification theorem for df-nfc 2183. (Contributed by Mario Carneiro, 13-Oct-2016.)

Definitiondf-nfc 2183* Define the not-free predicate for classes. This is read " is not free in ". Not-free means that the value of cannot affect the value of , e.g., any occurrence of in is effectively bound by a quantifier or something that expands to one (such as "there exists at most one"). It is defined in terms of the not-free predicate df-nf 1366 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfci 2184* Deduce that a class does not have free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcii 2185* Deduce that a class does not have free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcr 2186* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcrii 2187* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcri 2188* Consequence of the not-free predicate. (Note that unlike nfcr 2186, this does not require and to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcd 2189* Deduce that a class does not have free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfceqi 2190 Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcxfr 2191 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcxfrd 2192 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfceqdf 2193 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremnfcv 2194* If is disjoint from , then is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcvd 2195* If is disjoint from , then is not free in . (Contributed by Mario Carneiro, 7-Oct-2016.)

Theoremnfab1 2196 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfnfc1 2197 is bound in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfab 2198 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfaba1 2199 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremnfnfc 2200 Hypothesis builder for . (Contributed by Mario Carneiro, 11-Aug-2016.)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10511
 Copyright terms: Public domain < Previous  Next >