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

Theoremeleq2 2101 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Theoremeleq12 2102 Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))

Theoremeleq1i 2103 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)

Theoremeleq2i 2104 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)

Theoremeleq12i 2105 Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶𝐵𝐷)

Theoremeleq1d 2106 Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))

Theoremeleq2d 2107 Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))

Theoremeleq12d 2108 Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))

Theoremeleq1a 2109 A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
(𝐴𝐵 → (𝐶 = 𝐴𝐶𝐵))

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

Theoremeqeltrri 2111 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   𝐴𝐶       𝐵𝐶

Theoremeleqtri 2112 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶

Theoremeleqtrri 2113 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
𝐴𝐵    &   𝐶 = 𝐵       𝐴𝐶

Theoremeqeltrd 2114 Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremeqeltrrd 2115 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐶)       (𝜑𝐵𝐶)

Theoremeleqtrd 2116 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)

Theoremeleqtrrd 2117 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)

Theorem3eltr3i 2118 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐴𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐶𝐷

Theorem3eltr4i 2119 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐴𝐵    &   𝐶 = 𝐴    &   𝐷 = 𝐵       𝐶𝐷

Theorem3eltr3d 2120 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝐷)

Theorem3eltr4d 2121 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝐷)

Theorem3eltr3g 2122 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝐷)

Theorem3eltr4g 2123 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝐷)

Theoremsyl5eqel 2124 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
𝐴 = 𝐵    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremsyl5eqelr 2125 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
𝐵 = 𝐴    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremsyl5eleq 2126 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
𝐴𝐵    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)

Theoremsyl5eleqr 2127 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
𝐴𝐵    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)

Theoremsyl6eqel 2128 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐴 = 𝐵)    &   𝐵𝐶       (𝜑𝐴𝐶)

Theoremsyl6eqelr 2129 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐵 = 𝐴)    &   𝐵𝐶       (𝜑𝐴𝐶)

Theoremsyl6eleq 2130 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐴𝐵)    &   𝐵 = 𝐶       (𝜑𝐴𝐶)

Theoremsyl6eleqr 2131 A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐵       (𝜑𝐴𝐶)

Theoremeleq2s 2132 Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝐴𝐵𝜑)    &   𝐶 = 𝐵       (𝐴𝐶𝜑)

Theoremeqneltrd 2133 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 → ¬ 𝐵𝐶)       (𝜑 → ¬ 𝐴𝐶)

Theoremeqneltrrd 2134 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 → ¬ 𝐴𝐶)       (𝜑 → ¬ 𝐵𝐶)

Theoremneleqtrd 2135 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 → ¬ 𝐶𝐴)    &   (𝜑𝐴 = 𝐵)       (𝜑 → ¬ 𝐶𝐵)

Theoremneleqtrrd 2136 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 → ¬ 𝐶𝐵)    &   (𝜑𝐴 = 𝐵)       (𝜑 → ¬ 𝐶𝐴)

Theoremcleqh 2137* Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2201. (Contributed by NM, 5-Aug-1993.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)    &   (𝑦𝐵 → ∀𝑥 𝑦𝐵)       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Theoremnelneq 2138 A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
((𝐴𝐶 ∧ ¬ 𝐵𝐶) → ¬ 𝐴 = 𝐵)

Theoremnelneq2 2139 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵 = 𝐶)

Theoremeqsb3lem 2140* Lemma for eqsb3 2141. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)

Theoremeqsb3 2141* Substitution applied to an atomic wff (class version of equsb3 1825). (Contributed by Rodolfo Medina, 28-Apr-2010.)
([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)

Theoremclelsb3 2142* Substitution applied to an atomic wff (class version of elsb3 1852). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)

Theoremclelsb4 2143* Substitution applied to an atomic wff (class version of elsb4 1853). (Contributed by Jim Kingdon, 22-Nov-2018.)
([𝑥 / 𝑦]𝐴𝑦𝐴𝑥)

Theoremhbxfreq 2144 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1361 for equivalence version. (Contributed by NM, 21-Aug-2007.)
𝐴 = 𝐵    &   (𝑦𝐵 → ∀𝑥 𝑦𝐵)       (𝑦𝐴 → ∀𝑥 𝑦𝐴)

Theoremhblem 2145* Change the free variable of a hypothesis builder. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       (𝑧𝐴 → ∀𝑥 𝑧𝐴)

Theoremabeq2 2146* 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 2151 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 𝑥𝐴 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(𝑥𝐴𝜑))

Theoremabeq1 2147* Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))

Theoremabeq2i 2148 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.)
𝐴 = {𝑥𝜑}       (𝑥𝐴𝜑)

Theoremabeq1i 2149 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)
{𝑥𝜑} = 𝐴       (𝜑𝑥𝐴)

Theoremabeq2d 2150 Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
(𝜑𝐴 = {𝑥𝜓})       (𝜑 → (𝑥𝐴𝜓))

Theoremabbi 2151 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 2152* Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 5-Aug-1993.)
(𝑥𝐴𝜑)       𝐴 = {𝑥𝜑}

Theoremabbii 2153 Equivalent wff's yield equal class abstractions (inference rule). (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       {𝑥𝜑} = {𝑥𝜓}

Theoremabbid 2154 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 2155* Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 10-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})

Theoremabbi2dv 2156* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
(𝜑 → (𝑥𝐴𝜓))       (𝜑𝐴 = {𝑥𝜓})

Theoremabbi1dv 2157* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
(𝜑 → (𝜓𝑥𝐴))       (𝜑 → {𝑥𝜓} = 𝐴)

Theoremabid2 2158* A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)
{𝑥𝑥𝐴} = 𝐴

Theoremsb8ab 2159 Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)
𝑦𝜑       {𝑥𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑}

Theoremcbvab 2160 Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}

Theoremcbvabv 2161* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)
(𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}

Theoremclelab 2162* Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
(𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑))

Theoremclabel 2163* Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))

Theoremsbab 2164* 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]𝑧𝐴})

2.1.3  Class form not-free predicate

Syntaxwnfc 2165 Extend wff definition to include the not-free predicate for classes.
wff 𝑥𝐴

Theoremnfcjust 2166* Justification theorem for df-nfc 2167. (Contributed by Mario Carneiro, 13-Oct-2016.)
(∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)

Definitiondf-nfc 2167* 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 1350 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)

Theoremnfci 2168* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥 𝑦𝐴       𝑥𝐴

Theoremnfcii 2169* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       𝑥𝐴

Theoremnfcr 2170* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑥𝐴 → Ⅎ𝑥 𝑦𝐴)

Theoremnfcrii 2171* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴       (𝑦𝐴 → ∀𝑥 𝑦𝐴)

Theoremnfcri 2172* Consequence of the not-free predicate. (Note that unlike nfcr 2170, this does not require 𝑦 and 𝐴 to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴       𝑥 𝑦𝐴

Theoremnfcd 2173* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥 𝑦𝐴)       (𝜑𝑥𝐴)

Theoremnfceqi 2174 Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝐴 = 𝐵       (𝑥𝐴𝑥𝐵)

Theoremnfcxfr 2175 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 2176 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 2177 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴𝑥𝐵))

Theoremnfcv 2178* If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴

Theoremnfcvd 2179* If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)

Theoremnfab1 2180 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥{𝑥𝜑}

Theoremnfnfc1 2181 𝑥 is bound in 𝑥𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝑥𝐴

Theoremnfab 2182 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       𝑥{𝑦𝜑}

Theoremnfaba1 2183 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑥{𝑦 ∣ ∀𝑥𝜑}

Theoremnfnfc 2184 Hypothesis builder for 𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴       𝑥𝑦𝐴

Theoremnfeq 2185 Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴 = 𝐵

Theoremnfel 2186 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵

Theoremnfeq1 2187* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴       𝑥 𝐴 = 𝐵

Theoremnfel1 2188* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴       𝑥 𝐴𝐵

Theoremnfeq2 2189* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐵       𝑥 𝐴 = 𝐵

Theoremnfel2 2190* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐵       𝑥 𝐴𝐵

Theoremnfcrd 2191* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑𝑥𝐴)       (𝜑 → Ⅎ𝑥 𝑦𝐴)

Theoremnfeqd 2192 Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)

Theoremnfeld 2193 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝐵)

Theoremdrnfc1 2194 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 2195 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 2196 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥{𝑦𝜓})

Theoremdvelimdc 2197 Deduction form of dvelimc 2198. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑥𝜑    &   𝑧𝜑    &   (𝜑𝑥𝐴)    &   (𝜑𝑧𝐵)    &   (𝜑 → (𝑧 = 𝑦𝐴 = 𝐵))       (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵))

Theoremdvelimc 2198 Version of dvelim 1893 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑥𝐴    &   𝑧𝐵    &   (𝑧 = 𝑦𝐴 = 𝐵)       (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵)

Theoremnfcvf 2199 If 𝑥 and 𝑦 are distinct, then 𝑥 is not free in 𝑦. (Contributed by Mario Carneiro, 8-Oct-2016.)
(¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)

Theoremnfcvf2 2200 If 𝑥 and 𝑦 are distinct, then 𝑦 is not free in 𝑥. (Contributed by Mario Carneiro, 5-Dec-2016.)
(¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)

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