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Theorem List for Intuitionistic Logic Explorer - 1901-2000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsyl6req 1901 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
   &     C   =>     C
 
Theoremsyl6eqr 1902 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
   &     C    =>     C
 
Theoremsyl6reqr 1903 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
   &     C    =>     C
 
Theoremsylan9eq 1904 An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
   &     C   =>     C
 
Theoremsylan9req 1905 An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)
   &     C   =>     C
 
Theoremsylan9eqr 1906 An equality transitivity deduction. (Contributed by NM, 8-May-1994.)
   &     C   =>     C
 
Theorem3eqtr3g 1907 A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)
   &     C   &     D   =>     C  D
 
Theorem3eqtr3a 1908 A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)
   &     C   &     D   =>     C  D
 
Theorem3eqtr4g 1909 A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.)
   &     C    &     D    =>     C  D
 
Theorem3eqtr4a 1910 A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
   &     C    &     D    =>     C  D
 
Theoremeq2tri 1911 A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)
 C  D  F   &     D  C  G   =>     C  F  D  G
 
Theoremeleq1 1912 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
 C  C
 
Theoremeleq2 1913 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
 C  C
 
Theoremeleq12 1914 Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
 C  D  C  D
 
Theoremeleq1i 1915 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
   =>     C  C
 
Theoremeleq2i 1916 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
   =>     C  C
 
Theoremeleq12i 1917 Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
   &     C  D   =>     C  D
 
Theoremeleq1d 1918 Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
   =>     C  C
 
Theoremeleq2d 1919 Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.)
   =>     C  C
 
Theoremeleq12d 1920 Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
   &     C  D   =>     C  D
 
Theoremeleq1a 1921 A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
 C  C
 
Theoremeqeltri 1922 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
   &     C   =>     C
 
Theoremeqeltrri 1923 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
   &     C   =>     C
 
Theoremeleqtri 1924 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
   &     C   =>     C
 
Theoremeleqtrri 1925 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
   &     C    =>     C
 
Theoremeqeltrd 1926 Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)
   &     C   =>     C
 
Theoremeqeltrrd 1927 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
   &     C   =>     C
 
Theoremeleqtrd 1928 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
   &     C   =>     C
 
Theoremeleqtrrd 1929 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
   &     C    =>     C
 
Theorem3eltr3i 1930 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
   &     C   &     D   =>     C  D
 
Theorem3eltr4i 1931 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
   &     C    &     D    =>     C  D
 
Theorem3eltr3d 1932 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
   &     C   &     D   =>     C  D
 
Theorem3eltr4d 1933 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
   &     C    &     D    =>     C  D
 
Theorem3eltr3g 1934 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
   &     C   &     D   =>     C  D
 
Theorem3eltr4g 1935 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
   &     C    &     D    =>     C  D
 
Theoremsyl5eqel 1936 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
   &     C   =>     C
 
Theoremsyl5eqelr 1937 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
   &     C   =>     C
 
Theoremsyl5eleq 1938 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
   &     C   =>     C
 
Theoremsyl5eleqr 1939 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
   &     C    =>     C
 
Theoremsyl6eqel 1940 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
   &     C   =>     C
 
Theoremsyl6eqelr 1941 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
   &     C   =>     C
 
Theoremsyl6eleq 1942 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
   &     C   =>     C
 
Theoremsyl6eleqr 1943 A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
   &     C    =>     C
 
Theoremeleq2s 1944 Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
   &     C    =>     C
 
Theoremeqneltrd 1945 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.)
   &     C   =>     C
 
Theoremeqneltrrd 1946 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.)
   &     C   =>     C
 
Theoremneleqtrd 1947 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.)
 C    &       =>     C
 
Theoremneleqtrrd 1948 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.)
 C    &       =>     C
 
Theoremcleqh 1949* Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 5-Aug-1993.)
   &       =>   
 
Theoremnelneq 1950 A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
 C  C
 
Theoremnelneq2 1951 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
 C  C
 
Theoremeqsb3lem 1952* Lemma for eqsb3 1953. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 
Theoremeqsb3 1953* Substitution applied to an atomic wff (class version of equsb3 1724). (Contributed by Rodolfo Medina, 28-Apr-2010.)
 
Theoremclelsb3 1954* Substitution applied to an atomic wff (class version of elsb3 1751). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 
Theoremhbxfreq 1955 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1293 for equivalence version. (Contributed by NM, 21-Aug-2007.)
   &       =>   
 
Theoremhblem 1956* Change the free variable of a hypothesis builder. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)
   =>   
 
Theoremabeq2 1957* 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 1962 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 1958* Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
 {  | 
 }
 
Theoremabeq2i 1959 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.)
 {  |  }   =>   
 
Theoremabeq1i 1960 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)

 {  |  }    =>   
 
Theoremabeq2d 1961 Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
 {  |  }   =>   
 
Theoremabbi 1962 Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)
 {  |  }  {  |  }
 
Theoremabbi2i 1963* Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 5-Aug-1993.)
   =>     {  |  }
 
Theoremabbii 1964 Equivalent wff's yield equal class abstractions (inference rule). (Contributed by NM, 5-Aug-1993.)
   =>     {  |  }  {  |  }
 
Theoremabbid 1965 Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)

 F/   &       =>     {  |  }  {  |  }
 
Theoremabbidv 1966* Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 10-Aug-1993.)
   =>     {  |  }  {  |  }
 
Theoremabbi2dv 1967* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
   =>     {  |  }
 
Theoremabbi1dv 1968* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
   =>     {  |  }
 
Theoremabid2 1969* A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)

 {  |  }
 
Theoremcbvab 1970 Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)

 F/   &     F/   &       =>     {  |  }  {  |  }
 
Theoremcbvabv 1971* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)
   =>    
 {  |  }  {  |  }
 
Theoremclelab 1972* Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
 {  |  }
 
Theoremclabel 1973* Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
 {  | 
 }
 
Theoremsbab 1974* 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 1975 Extend wff definition to include the not-free predicate for classes.
 F/_
 
Theoremnfcjust 1976* Justification theorem for df-nfc 1977. (Contributed by Mario Carneiro, 13-Oct-2016.)
 F/  F/
 
Definitiondf-nfc 1977* 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 1282 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)
 F/_  F/
 
Theoremnfci 1978* Deduce that a class does not have free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)

 F/    =>     F/_
 
Theoremnfcii 1979* Deduce that a class does not have free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
   =>     F/_
 
Theoremnfcr 1980* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
 F/_  F/
 
Theoremnfcrii 1981* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
 F/_   =>   
 
Theoremnfcri 1982* Consequence of the not-free predicate. (Note that unlike nfcr 1980, this does not require and to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)
 F/_   =>    
 F/
 
Theoremnfcd 1983* Deduce that a class does not have free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)

 F/   &     F/    =>     F/_
 
Theoremnfceqi 1984 Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
   =>     F/_  F/_
 
Theoremnfcxfr 1985 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
   &     F/_   =>     F/_
 
Theoremnfcxfrd 1986 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
   &     F/_   =>     F/_
 
Theoremnfceqdf 1987 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)

 F/   &       =>     F/_  F/_
 
Theoremnfcv 1988* If is disjoint from , then is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)
 F/_
 
Theoremnfcvd 1989* If is disjoint from , then is not free in . (Contributed by Mario Carneiro, 7-Oct-2016.)
 F/_
 
Theoremnfab1 1990 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
 F/_ {  | 
 }
 
Theoremnfnfc1 1991 is bound in  F/_. (Contributed by Mario Carneiro, 11-Aug-2016.)

 F/ F/_
 
Theoremnfab 1992 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)

 F/   =>     F/_ {  | 
 }
 
Theoremnfaba1 1993 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
 F/_ {  |  }
 
Theoremnfnfc 1994 Hypothesis builder for  F/_. (Contributed by Mario Carneiro, 11-Aug-2016.)
 F/_   =>    
 F/ F/_
 
Theoremnfeq 1995 Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
 F/_   &     F/_   =>     F/
 
Theoremnfel 1996 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
 F/_   &     F/_   =>     F/
 
Theoremnfeq1 1997* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 F/_   =>    
 F/
 
Theoremnfel1 1998* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 F/_   =>    
 F/
 
Theoremnfeq2 1999* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 F/_   =>    
 F/
 
Theoremnfel2 2000* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
 F/_   =>    
 F/
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