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

Theoremeleq2i 2101 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
A = B       (𝐶 A𝐶 B)

Theoremeleq12i 2102 Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
A = B    &   𝐶 = 𝐷       (A 𝐶B 𝐷)

Theoremeleq1d 2103 Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(φA = B)       (φ → (A 𝐶B 𝐶))

Theoremeleq2d 2104 Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.)
(φA = B)       (φ → (𝐶 A𝐶 B))

Theoremeleq12d 2105 Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A 𝐶B 𝐷))

Theoremeleq1a 2106 A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
(A B → (𝐶 = A𝐶 B))

Theoremeqeltri 2107 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
A = B    &   B 𝐶       A 𝐶

Theoremeqeltrri 2108 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
A = B    &   A 𝐶       B 𝐶

Theoremeleqtri 2109 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
A B    &   B = 𝐶       A 𝐶

Theoremeleqtrri 2110 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
A B    &   𝐶 = B       A 𝐶

Theoremeqeltrd 2111 Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)
(φA = B)    &   (φB 𝐶)       (φA 𝐶)

Theoremeqeltrrd 2112 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(φA = B)    &   (φA 𝐶)       (φB 𝐶)

Theoremeleqtrd 2113 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(φA B)    &   (φB = 𝐶)       (φA 𝐶)

Theoremeleqtrrd 2114 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(φA B)    &   (φ𝐶 = B)       (φA 𝐶)

Theorem3eltr3i 2115 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
A B    &   A = 𝐶    &   B = 𝐷       𝐶 𝐷

Theorem3eltr4i 2116 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
A B    &   𝐶 = A    &   𝐷 = B       𝐶 𝐷

Theorem3eltr3d 2117 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(φA B)    &   (φA = 𝐶)    &   (φB = 𝐷)       (φ𝐶 𝐷)

Theorem3eltr4d 2118 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(φA B)    &   (φ𝐶 = A)    &   (φ𝐷 = B)       (φ𝐶 𝐷)

Theorem3eltr3g 2119 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(φA B)    &   A = 𝐶    &   B = 𝐷       (φ𝐶 𝐷)

Theorem3eltr4g 2120 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(φA B)    &   𝐶 = A    &   𝐷 = B       (φ𝐶 𝐷)

Theoremsyl5eqel 2121 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
A = B    &   (φB 𝐶)       (φA 𝐶)

Theoremsyl5eqelr 2122 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
B = A    &   (φB 𝐶)       (φA 𝐶)

Theoremsyl5eleq 2123 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
A B    &   (φB = 𝐶)       (φA 𝐶)

Theoremsyl5eleqr 2124 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
A B    &   (φ𝐶 = B)       (φA 𝐶)

Theoremsyl6eqel 2125 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(φA = B)    &   B 𝐶       (φA 𝐶)

Theoremsyl6eqelr 2126 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(φB = A)    &   B 𝐶       (φA 𝐶)

Theoremsyl6eleq 2127 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(φA B)    &   B = 𝐶       (φA 𝐶)

Theoremsyl6eleqr 2128 A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
(φA B)    &   𝐶 = B       (φA 𝐶)

Theoremeleq2s 2129 Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(A Bφ)    &   𝐶 = B       (A 𝐶φ)

Theoremeqneltrd 2130 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.)
(φA = B)    &   (φ → ¬ B 𝐶)       (φ → ¬ A 𝐶)

Theoremeqneltrrd 2131 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.)
(φA = B)    &   (φ → ¬ A 𝐶)       (φ → ¬ B 𝐶)

Theoremneleqtrd 2132 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.)
(φ → ¬ 𝐶 A)    &   (φA = B)       (φ → ¬ 𝐶 B)

Theoremneleqtrrd 2133 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.)
(φ → ¬ 𝐶 B)    &   (φA = B)       (φ → ¬ 𝐶 A)

Theoremcleqh 2134* Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2198. (Contributed by NM, 5-Aug-1993.)
(y Ax y A)    &   (y Bx y B)       (A = Bx(x Ax B))

Theoremnelneq 2135 A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
((A 𝐶 ¬ B 𝐶) → ¬ A = B)

Theoremnelneq2 2136 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
((A B ¬ A 𝐶) → ¬ B = 𝐶)

Theoremeqsb3lem 2137* Lemma for eqsb3 2138. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([x / y]y = Ax = A)

Theoremeqsb3 2138* Substitution applied to an atomic wff (class version of equsb3 1822). (Contributed by Rodolfo Medina, 28-Apr-2010.)
([x / y]y = Ax = A)

Theoremclelsb3 2139* Substitution applied to an atomic wff (class version of elsb3 1849). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([x / y]y Ax A)

Theoremclelsb4 2140* Substitution applied to an atomic wff (class version of elsb4 1850). (Contributed by Jim Kingdon, 22-Nov-2018.)
([x / y]A yA x)

Theoremhbxfreq 2141 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1358 for equivalence version. (Contributed by NM, 21-Aug-2007.)
A = B    &   (y Bx y B)       (y Ax y A)

Theoremhblem 2142* Change the free variable of a hypothesis builder. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)
(y Ax y A)       (z Ax z A)

Theoremabeq2 2143* 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 2148 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 φ (that has a free variable x) to a theorem with a class variable A, we substitute x A for φ throughout and simplify, where A is a new class variable not already in the wff. Conversely, to convert a theorem with a class variable A to one with φ, we substitute {xφ} for A throughout and simplify, where x and φ 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.)

(A = {xφ} ↔ x(x Aφ))

Theoremabeq1 2144* Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
({xφ} = Ax(φx A))

Theoremabeq2i 2145 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.)
A = {xφ}       (x Aφ)

Theoremabeq1i 2146 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)
{xφ} = A       (φx A)

Theoremabeq2d 2147 Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
(φA = {xψ})       (φ → (x Aψ))

Theoremabbi 2148 Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)
(x(φψ) ↔ {xφ} = {xψ})

Theoremabbi2i 2149* Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 5-Aug-1993.)
(x Aφ)       A = {xφ}

Theoremabbii 2150 Equivalent wff's yield equal class abstractions (inference rule). (Contributed by NM, 5-Aug-1993.)
(φψ)       {xφ} = {xψ}

Theoremabbid 2151 Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
xφ    &   (φ → (ψχ))       (φ → {xψ} = {xχ})

Theoremabbidv 2152* Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 10-Aug-1993.)
(φ → (ψχ))       (φ → {xψ} = {xχ})

Theoremabbi2dv 2153* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
(φ → (x Aψ))       (φA = {xψ})

Theoremabbi1dv 2154* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
(φ → (ψx A))       (φ → {xψ} = A)

Theoremabid2 2155* A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)
{xx A} = A

Theoremsb8ab 2156 Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)
yφ       {xφ} = {y ∣ [y / x]φ}

Theoremcbvab 2157 Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)
yφ    &   xψ    &   (x = y → (φψ))       {xφ} = {yψ}

Theoremcbvabv 2158* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)
(x = y → (φψ))       {xφ} = {yψ}

Theoremclelab 2159* Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
(A {xφ} ↔ x(x = A φ))

Theoremclabel 2160* Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
({xφ} Ay(y A x(x yφ)))

Theoremsbab 2161* The right-hand side of the second equality is a way of representing proper substitution of y for x into a class variable. (Contributed by NM, 14-Sep-2003.)
(x = yA = {z ∣ [y / x]z A})

2.1.3  Class form not-free predicate

Syntaxwnfc 2162 Extend wff definition to include the not-free predicate for classes.
wff xA

Theoremnfcjust 2163* Justification theorem for df-nfc 2164. (Contributed by Mario Carneiro, 13-Oct-2016.)
(yx y Azx z A)

Definitiondf-nfc 2164* Define the not-free predicate for classes. This is read "x is not free in A". Not-free means that the value of x cannot affect the value of A, e.g., any occurrence of x in A 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 1347 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)
(xAyx y A)

Theoremnfci 2165* Deduce that a class A does not have x free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
x y A       xA

Theoremnfcii 2166* Deduce that a class A does not have x free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
(y Ax y A)       xA

Theoremnfcr 2167* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(xA → Ⅎx y A)

Theoremnfcrii 2168* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
xA       (y Ax y A)

Theoremnfcri 2169* Consequence of the not-free predicate. (Note that unlike nfcr 2167, this does not require y and A to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)
xA       x y A

Theoremnfcd 2170* Deduce that a class A does not have x free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
yφ    &   (φ → Ⅎx y A)       (φxA)

Theoremnfceqi 2171 Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
A = B       (xAxB)

Theoremnfcxfr 2172 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
A = B    &   xB       xA

Theoremnfcxfrd 2173 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
A = B    &   (φxB)       (φxA)

Theoremnfceqdf 2174 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)
xφ    &   (φA = B)       (φ → (xAxB))

Theoremnfcv 2175* If x is disjoint from A, then x is not free in A. (Contributed by Mario Carneiro, 11-Aug-2016.)
xA

Theoremnfcvd 2176* If x is disjoint from A, then x is not free in A. (Contributed by Mario Carneiro, 7-Oct-2016.)
(φxA)

Theoremnfab1 2177 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
x{xφ}

Theoremnfnfc1 2178 x is bound in xA. (Contributed by Mario Carneiro, 11-Aug-2016.)
xxA

Theoremnfab 2179 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
xφ       x{yφ}

Theoremnfaba1 2180 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
x{yxφ}

Theoremnfnfc 2181 Hypothesis builder for yA. (Contributed by Mario Carneiro, 11-Aug-2016.)
xA       xyA

Theoremnfeq 2182 Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
xA    &   xB       x A = B

Theoremnfel 2183 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
xA    &   xB       x A B

Theoremnfeq1 2184* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
xA       x A = B

Theoremnfel1 2185* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
xA       x A B

Theoremnfeq2 2186* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
xB       x A = B

Theoremnfel2 2187* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
xB       x A B

Theoremnfcrd 2188* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(φxA)       (φ → Ⅎx y A)

Theoremnfeqd 2189 Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
(φxA)    &   (φxB)       (φ → Ⅎx A = B)

Theoremnfeld 2190 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
(φxA)    &   (φxB)       (φ → Ⅎx A B)

Theoremdrnfc1 2191 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
(x x = yA = B)       (x x = y → (xAyB))

Theoremdrnfc2 2192 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
(x x = yA = B)       (x x = y → (zAzB))

Theoremnfabd 2193 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
yφ    &   (φ → Ⅎxψ)       (φx{yψ})

Theoremdvelimdc 2194 Deduction form of dvelimc 2195. (Contributed by Mario Carneiro, 8-Oct-2016.)
xφ    &   zφ    &   (φxA)    &   (φzB)    &   (φ → (z = yA = B))       (φ → (¬ x x = yxB))

Theoremdvelimc 2195 Version of dvelim 1890 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)
xA    &   zB    &   (z = yA = B)       x x = yxB)

Theoremnfcvf 2196 If x and y are distinct, then x is not free in y. (Contributed by Mario Carneiro, 8-Oct-2016.)
x x = yxy)

Theoremnfcvf2 2197 If x and y are distinct, then y is not free in x. (Contributed by Mario Carneiro, 5-Dec-2016.)
x x = yyx)

Theoremcleqf 2198 Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2134. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
xA    &   xB       (A = Bx(x Ax B))

Theoremabid2f 2199 A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
xA       {xx A} = A

Theoremsbabel 2200* Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.)
xA       ([y / x]{zφ} A ↔ {z ∣ [y / x]φ} A)

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