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

Theorem3eltr4g 2101 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(φA B)    &   𝐶 = A    &   𝐷 = B       (φ𝐶 𝐷)

Theoremsyl5eqel 2102 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
A = B    &   (φB 𝐶)       (φA 𝐶)

Theoremsyl5eqelr 2103 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
B = A    &   (φB 𝐶)       (φA 𝐶)

Theoremsyl5eleq 2104 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
A B    &   (φB = 𝐶)       (φA 𝐶)

Theoremsyl5eleqr 2105 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
A B    &   (φ𝐶 = B)       (φA 𝐶)

Theoremsyl6eqel 2106 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(φA = B)    &   B 𝐶       (φA 𝐶)

Theoremsyl6eqelr 2107 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(φB = A)    &   B 𝐶       (φA 𝐶)

Theoremsyl6eleq 2108 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(φA B)    &   B = 𝐶       (φA 𝐶)

Theoremsyl6eleqr 2109 A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
(φA B)    &   𝐶 = B       (φA 𝐶)

Theoremeleq2s 2110 Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(A Bφ)    &   𝐶 = B       (A 𝐶φ)

Theoremeqneltrd 2111 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 2112 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 2113 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 2114 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 2115* Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2179. (Contributed by NM, 5-Aug-1993.)
(y Ax y A)    &   (y Bx y B)       (A = Bx(x Ax B))

Theoremnelneq 2116 A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
((A 𝐶 ¬ B 𝐶) → ¬ A = B)

Theoremnelneq2 2117 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
((A B ¬ A 𝐶) → ¬ B = 𝐶)

Theoremeqsb3lem 2118* Lemma for eqsb3 2119. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([x / y]y = Ax = A)

Theoremeqsb3 2119* Substitution applied to an atomic wff (class version of equsb3 1803). (Contributed by Rodolfo Medina, 28-Apr-2010.)
([x / y]y = Ax = A)

Theoremclelsb3 2120* Substitution applied to an atomic wff (class version of elsb3 1830). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([x / y]y Ax A)

Theoremclelsb4 2121* Substitution applied to an atomic wff (class version of elsb4 1831). (Contributed by Jim Kingdon, 22-Nov-2018.)
([x / y]A yA x)

Theoremhbxfreq 2122 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1337 for equivalence version. (Contributed by NM, 21-Aug-2007.)
A = B    &   (y Bx y B)       (y Ax y A)

Theoremhblem 2123* 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 2124* 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 2129 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 2125* Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
({xφ} = Ax(φx A))

Theoremabeq2i 2126 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.)
A = {xφ}       (x Aφ)

Theoremabeq1i 2127 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)
{xφ} = A       (φx A)

Theoremabeq2d 2128 Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
(φA = {xψ})       (φ → (x Aψ))

Theoremabbi 2129 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 2130* Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 5-Aug-1993.)
(x Aφ)       A = {xφ}

Theoremabbii 2131 Equivalent wff's yield equal class abstractions (inference rule). (Contributed by NM, 5-Aug-1993.)
(φψ)       {xφ} = {xψ}

Theoremabbid 2132 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 2133* Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 10-Aug-1993.)
(φ → (ψχ))       (φ → {xψ} = {xχ})

Theoremabbi2dv 2134* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
(φ → (x Aψ))       (φA = {xψ})

Theoremabbi1dv 2135* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
(φ → (ψx A))       (φ → {xψ} = A)

Theoremabid2 2136* A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)
{xx A} = A

Theoremsb8ab 2137 Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)
yφ       {xφ} = {y ∣ [y / x]φ}

Theoremcbvab 2138 Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)
yφ    &   xψ    &   (x = y → (φψ))       {xφ} = {yψ}

Theoremcbvabv 2139* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)
(x = y → (φψ))       {xφ} = {yψ}

Theoremclelab 2140* Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
(A {xφ} ↔ x(x = A φ))

Theoremclabel 2141* Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
({xφ} Ay(y A x(x yφ)))

Theoremsbab 2142* 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 2143 Extend wff definition to include the not-free predicate for classes.
wff xA

Theoremnfcjust 2144* Justification theorem for df-nfc 2145. (Contributed by Mario Carneiro, 13-Oct-2016.)
(yx y Azx z A)

Definitiondf-nfc 2145* 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 1326 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)
(xAyx y A)

Theoremnfci 2146* Deduce that a class A does not have x free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
x y A       xA

Theoremnfcii 2147* 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 2148* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(xA → Ⅎx y A)

Theoremnfcrii 2149* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
xA       (y Ax y A)

Theoremnfcri 2150* Consequence of the not-free predicate. (Note that unlike nfcr 2148, this does not require y and A to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)
xA       x y A

Theoremnfcd 2151* 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 2152 Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
A = B       (xAxB)

Theoremnfcxfr 2153 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 2154 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 2155 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)
xφ    &   (φA = B)       (φ → (xAxB))

Theoremnfcv 2156* If x is disjoint from A, then x is not free in A. (Contributed by Mario Carneiro, 11-Aug-2016.)
xA

Theoremnfcvd 2157* If x is disjoint from A, then x is not free in A. (Contributed by Mario Carneiro, 7-Oct-2016.)
(φxA)

Theoremnfab1 2158 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
x{xφ}

Theoremnfnfc1 2159 x is bound in xA. (Contributed by Mario Carneiro, 11-Aug-2016.)
xxA

Theoremnfab 2160 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
xφ       x{yφ}

Theoremnfaba1 2161 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
x{yxφ}

Theoremnfnfc 2162 Hypothesis builder for yA. (Contributed by Mario Carneiro, 11-Aug-2016.)
xA       xyA

Theoremnfeq 2163 Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
xA    &   xB       x A = B

Theoremnfel 2164 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
xA    &   xB       x A B

Theoremnfeq1 2165* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
xA       x A = B

Theoremnfel1 2166* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
xA       x A B

Theoremnfeq2 2167* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
xB       x A = B

Theoremnfel2 2168* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
xB       x A B

Theoremnfcrd 2169* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(φxA)       (φ → Ⅎx y A)

Theoremnfeqd 2170 Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
(φxA)    &   (φxB)       (φ → Ⅎx A = B)

Theoremnfeld 2171 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
(φxA)    &   (φxB)       (φ → Ⅎx A B)

Theoremdrnfc1 2172 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 2173 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 2174 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
yφ    &   (φ → Ⅎxψ)       (φx{yψ})

Theoremdvelimdc 2175 Deduction form of dvelimc 2176. (Contributed by Mario Carneiro, 8-Oct-2016.)
xφ    &   zφ    &   (φxA)    &   (φzB)    &   (φ → (z = yA = B))       (φ → (¬ x x = yxB))

Theoremdvelimc 2176 Version of dvelim 1871 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)
xA    &   zB    &   (z = yA = B)       x x = yxB)

Theoremnfcvf 2177 If x and y are distinct, then x is not free in y. (Contributed by Mario Carneiro, 8-Oct-2016.)
x x = yxy)

Theoremnfcvf2 2178 If x and y are distinct, then y is not free in x. (Contributed by Mario Carneiro, 5-Dec-2016.)
x x = yyx)

Theoremcleqf 2179 Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2115. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
xA    &   xB       (A = Bx(x Ax B))

Theoremabid2f 2180 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 2181* 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)

2.1.4  Negated equality and membership

Syntaxwne 2182 Extend wff notation to include inequality.
wff AB

Syntaxwnel 2183 Extend wff notation to include negated membership.
wff AB

Definitiondf-ne 2184 Define inequality. (Contributed by NM, 5-Aug-1993.)
(AB ↔ ¬ A = B)

Definitiondf-nel 2185 Define negated membership. (Contributed by NM, 7-Aug-1994.)
(AB ↔ ¬ A B)

2.1.4.1  Negated equality

Theoremneii 2186 Inference associated with df-ne 2184. (Contributed by BJ, 7-Jul-2018.)
AB        ¬ A = B

Theoremneir 2187 Inference associated with df-ne 2184. (Contributed by BJ, 7-Jul-2018.)
¬ A = B       AB

Theoremnner 2188 Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)
(A = B → ¬ AB)

Theoremnnedc 2189 Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
(DECID A = B → (¬ ABA = B))

Theoremneirr 2190 No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
¬ AA

Theoremexmidnedc 2191 Excluded middle with equality and inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
(DECID A = B → (A = B AB))

Theoremnonconne 2192 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
¬ (A = B AB)

Theoremneeq1 2193 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
(A = B → (A𝐶B𝐶))

Theoremneeq2 2194 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
(A = B → (𝐶A𝐶B))

Theoremneeq1i 2195 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
A = B       (A𝐶B𝐶)

Theoremneeq2i 2196 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
A = B       (𝐶A𝐶B)

Theoremneeq12i 2197 Inference for inequality. (Contributed by NM, 24-Jul-2012.)
A = B    &   𝐶 = 𝐷       (A𝐶B𝐷)

Theoremneeq1d 2198 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
(φA = B)       (φ → (A𝐶B𝐶))

Theoremneeq2d 2199 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
(φA = B)       (φ → (𝐶A𝐶B))

Theoremneeq12d 2200 Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶B𝐷))

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