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Theorem List for Intuitionistic Logic Explorer - 2201-2300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeleq2w 2201 Weaker version of eleq2 2203 (but more general than elequ2 1691) not depending on ax-ext 2121 nor df-cleq 2132. (Contributed by BJ, 29-Sep-2019.)
(𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
 
Theoremeleq1 2202 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremeleq2 2203 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremeleq12 2204 Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
 
Theoremeleq1i 2205 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)
 
Theoremeleq2i 2206 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)
 
Theoremeleq12i 2207 Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶𝐵𝐷)
 
Theoremeleq1d 2208 Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))
 
Theoremeleq2d 2209 Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))
 
Theoremeleq12d 2210 Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))
 
Theoremeleq1a 2211 A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
(𝐴𝐵 → (𝐶 = 𝐴𝐶𝐵))
 
Theoremeqeltri 2212 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   𝐵𝐶       𝐴𝐶
 
Theoremeqeltrri 2213 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   𝐴𝐶       𝐵𝐶
 
Theoremeleqtri 2214 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶
 
Theoremeleqtrri 2215 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
𝐴𝐵    &   𝐶 = 𝐵       𝐴𝐶
 
Theoremeqeltrd 2216 Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremeqeltrrd 2217 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐶)       (𝜑𝐵𝐶)
 
Theoremeleqtrd 2218 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)
 
Theoremeleqtrrd 2219 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)
 
Theorem3eltr3i 2220 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐴𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐶𝐷
 
Theorem3eltr4i 2221 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐴𝐵    &   𝐶 = 𝐴    &   𝐷 = 𝐵       𝐶𝐷
 
Theorem3eltr3d 2222 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝐷)
 
Theorem3eltr4d 2223 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝐷)
 
Theorem3eltr3g 2224 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝐷)
 
Theorem3eltr4g 2225 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝐷)
 
Theoremeqeltrid 2226 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
𝐴 = 𝐵    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremeqeltrrid 2227 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
𝐵 = 𝐴    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremeleqtrid 2228 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
𝐴𝐵    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)
 
Theoremeleqtrrid 2229 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
𝐴𝐵    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)
 
Theoremeqeltrdi 2230 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐴 = 𝐵)    &   𝐵𝐶       (𝜑𝐴𝐶)
 
Theoremeqeltrrdi 2231 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐵 = 𝐴)    &   𝐵𝐶       (𝜑𝐴𝐶)
 
Theoremeleqtrdi 2232 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐴𝐵)    &   𝐵 = 𝐶       (𝜑𝐴𝐶)
 
Theoremeleqtrrdi 2233 A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐵       (𝜑𝐴𝐶)
 
Theoremeleq2s 2234 Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝐴𝐵𝜑)    &   𝐶 = 𝐵       (𝐴𝐶𝜑)
 
Theoremeqneltrd 2235 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 2236 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 2237 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 2238 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 2239* Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2305. (Contributed by NM, 5-Aug-1993.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)    &   (𝑦𝐵 → ∀𝑥 𝑦𝐵)       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremnelneq 2240 A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
((𝐴𝐶 ∧ ¬ 𝐵𝐶) → ¬ 𝐴 = 𝐵)
 
Theoremnelneq2 2241 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵 = 𝐶)
 
Theoremeqsb3lem 2242* Lemma for eqsb3 2243. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)
 
Theoremeqsb3 2243* Substitution applied to an atomic wff (class version of equsb3 1924). (Contributed by Rodolfo Medina, 28-Apr-2010.)
([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)
 
Theoremclelsb3 2244* Substitution applied to an atomic wff (class version of elsb3 1951). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
 
Theoremclelsb4 2245* Substitution applied to an atomic wff (class version of elsb4 1952). (Contributed by Jim Kingdon, 22-Nov-2018.)
([𝑦 / 𝑥]𝐴𝑥𝐴𝑦)
 
Theoremhbxfreq 2246 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1448 for equivalence version. (Contributed by NM, 21-Aug-2007.)
𝐴 = 𝐵    &   (𝑦𝐵 → ∀𝑥 𝑦𝐵)       (𝑦𝐴 → ∀𝑥 𝑦𝐴)
 
Theoremhblem 2247* 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 2248* 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 2253 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 2249* Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
 
Theoremabeq2i 2250 Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 3-Apr-1996.)
𝐴 = {𝑥𝜑}       (𝑥𝐴𝜑)
 
Theoremabeq1i 2251 Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 31-Jul-1994.)
{𝑥𝜑} = 𝐴       (𝜑𝑥𝐴)
 
Theoremabeq2d 2252 Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
(𝜑𝐴 = {𝑥𝜓})       (𝜑 → (𝑥𝐴𝜓))
 
Theoremabbi 2253 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 2254* Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 5-Aug-1993.)
(𝑥𝐴𝜑)       𝐴 = {𝑥𝜑}
 
Theoremabbii 2255 Equivalent wff's yield equal class abstractions (inference form). (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       {𝑥𝜑} = {𝑥𝜓}
 
Theoremabbid 2256 Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})
 
Theoremabbidv 2257* Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 10-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})
 
Theoremabbi2dv 2258* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
(𝜑 → (𝑥𝐴𝜓))       (𝜑𝐴 = {𝑥𝜓})
 
Theoremabbi1dv 2259* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
(𝜑 → (𝜓𝑥𝐴))       (𝜑 → {𝑥𝜓} = 𝐴)
 
Theoremabid2 2260* A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)
{𝑥𝑥𝐴} = 𝐴
 
Theoremsb8ab 2261 Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)
𝑦𝜑       {𝑥𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑}
 
Theoremcbvabw 2262* Version of cbvab 2263 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}
 
Theoremcbvab 2263 Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}
 
Theoremcbvabv 2264* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)
(𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}
 
Theoremclelab 2265* Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
(𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑))
 
Theoremclabel 2266* Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))
 
Theoremsbab 2267* 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 2268 Extend wff definition to include the not-free predicate for classes.
wff 𝑥𝐴
 
Theoremnfcjust 2269* Justification theorem for df-nfc 2270. (Contributed by Mario Carneiro, 13-Oct-2016.)
(∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
 
Definitiondf-nfc 2270* 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 1437 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
 
Theoremnfci 2271* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥 𝑦𝐴       𝑥𝐴
 
Theoremnfcii 2272* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       𝑥𝐴
 
Theoremnfcr 2273* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑥𝐴 → Ⅎ𝑥 𝑦𝐴)
 
Theoremnfcrii 2274* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴       (𝑦𝐴 → ∀𝑥 𝑦𝐴)
 
Theoremnfcri 2275* Consequence of the not-free predicate. (Note that unlike nfcr 2273, this does not require 𝑦 and 𝐴 to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴       𝑥 𝑦𝐴
 
Theoremnfcd 2276* 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 2277 Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝐴 = 𝐵       (𝑥𝐴𝑥𝐵)
 
Theoremnfcxfr 2278 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 2279 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 2280 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴𝑥𝐵))
 
Theoremnfcv 2281* If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴
 
Theoremnfcvd 2282* If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)
 
Theoremnfab1 2283 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥{𝑥𝜑}
 
Theoremnfnfc1 2284 𝑥 is bound in 𝑥𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝑥𝐴
 
Theoremclelsb3f 2285 Substitution applied to an atomic wff (class version of elsb3 1951). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
𝑥𝐴       ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
 
Theoremnfab 2286 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       𝑥{𝑦𝜑}
 
Theoremnfaba1 2287 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑥{𝑦 ∣ ∀𝑥𝜑}
 
Theoremnfnfc 2288 Hypothesis builder for 𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴       𝑥𝑦𝐴
 
Theoremnfeq 2289 Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴 = 𝐵
 
Theoremnfel 2290 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵
 
Theoremnfeq1 2291* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴       𝑥 𝐴 = 𝐵
 
Theoremnfel1 2292* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴       𝑥 𝐴𝐵
 
Theoremnfeq2 2293* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐵       𝑥 𝐴 = 𝐵
 
Theoremnfel2 2294* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐵       𝑥 𝐴𝐵
 
Theoremnfcrd 2295* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑𝑥𝐴)       (𝜑 → Ⅎ𝑥 𝑦𝐴)
 
Theoremnfeqd 2296 Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)
 
Theoremnfeld 2297 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝐵)
 
Theoremdrnfc1 2298 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 2299 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 2300 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥{𝑦𝜓})
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