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Theorem List for Intuitionistic Logic Explorer - 2201-2300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremabbi2dv 2201* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
(𝜑 → (𝑥𝐴𝜓))       (𝜑𝐴 = {𝑥𝜓})

Theoremabbi1dv 2202* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
(𝜑 → (𝜓𝑥𝐴))       (𝜑 → {𝑥𝜓} = 𝐴)

Theoremabid2 2203* A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)
{𝑥𝑥𝐴} = 𝐴

Theoremsb8ab 2204 Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)
𝑦𝜑       {𝑥𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑}

Theoremcbvab 2205 Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}

Theoremcbvabv 2206* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)
(𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}

Theoremclelab 2207* Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
(𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑))

Theoremclabel 2208* Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))

Theoremsbab 2209* 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 2210 Extend wff definition to include the not-free predicate for classes.
wff 𝑥𝐴

Theoremnfcjust 2211* Justification theorem for df-nfc 2212. (Contributed by Mario Carneiro, 13-Oct-2016.)
(∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)

Definitiondf-nfc 2212* 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 1391 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)

Theoremnfci 2213* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥 𝑦𝐴       𝑥𝐴

Theoremnfcii 2214* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       𝑥𝐴

Theoremnfcr 2215* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑥𝐴 → Ⅎ𝑥 𝑦𝐴)

Theoremnfcrii 2216* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴       (𝑦𝐴 → ∀𝑥 𝑦𝐴)

Theoremnfcri 2217* Consequence of the not-free predicate. (Note that unlike nfcr 2215, this does not require 𝑦 and 𝐴 to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴       𝑥 𝑦𝐴

Theoremnfcd 2218* 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 2219 Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝐴 = 𝐵       (𝑥𝐴𝑥𝐵)

Theoremnfcxfr 2220 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 2221 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 2222 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴𝑥𝐵))

Theoremnfcv 2223* If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴

Theoremnfcvd 2224* If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)

Theoremnfab1 2225 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥{𝑥𝜑}

Theoremnfnfc1 2226 𝑥 is bound in 𝑥𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝑥𝐴

Theoremnfab 2227 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       𝑥{𝑦𝜑}

Theoremnfaba1 2228 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑥{𝑦 ∣ ∀𝑥𝜑}

Theoremnfnfc 2229 Hypothesis builder for 𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴       𝑥𝑦𝐴

Theoremnfeq 2230 Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴 = 𝐵

Theoremnfel 2231 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵

Theoremnfeq1 2232* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴       𝑥 𝐴 = 𝐵

Theoremnfel1 2233* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴       𝑥 𝐴𝐵

Theoremnfeq2 2234* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐵       𝑥 𝐴 = 𝐵

Theoremnfel2 2235* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐵       𝑥 𝐴𝐵

Theoremnfcrd 2236* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑𝑥𝐴)       (𝜑 → Ⅎ𝑥 𝑦𝐴)

Theoremnfeqd 2237 Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)

Theoremnfeld 2238 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝐵)

Theoremdrnfc1 2239 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 2240 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 2241 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥{𝑦𝜓})

Theoremdvelimdc 2242 Deduction form of dvelimc 2243. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑥𝜑    &   𝑧𝜑    &   (𝜑𝑥𝐴)    &   (𝜑𝑧𝐵)    &   (𝜑 → (𝑧 = 𝑦𝐴 = 𝐵))       (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵))

Theoremdvelimc 2243 Version of dvelim 1936 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑥𝐴    &   𝑧𝐵    &   (𝑧 = 𝑦𝐴 = 𝐵)       (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵)

Theoremnfcvf 2244 If 𝑥 and 𝑦 are distinct, then 𝑥 is not free in 𝑦. (Contributed by Mario Carneiro, 8-Oct-2016.)
(¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)

Theoremnfcvf2 2245 If 𝑥 and 𝑦 are distinct, then 𝑦 is not free in 𝑥. (Contributed by Mario Carneiro, 5-Dec-2016.)
(¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)

Theoremcleqf 2246 Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2182. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Theoremabid2f 2247 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       {𝑥𝑥𝐴} = 𝐴

Theoremsbabel 2248* 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

2.1.4.1  Negated equality

Syntaxwne 2249 Extend wff notation to include inequality.
wff 𝐴𝐵

Definitiondf-ne 2250 Define inequality. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)

Theoremneii 2251 Inference associated with df-ne 2250. (Contributed by BJ, 7-Jul-2018.)
𝐴𝐵        ¬ 𝐴 = 𝐵

Theoremneir 2252 Inference associated with df-ne 2250. (Contributed by BJ, 7-Jul-2018.)
¬ 𝐴 = 𝐵       𝐴𝐵

Theoremnner 2253 Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)
(𝐴 = 𝐵 → ¬ 𝐴𝐵)

Theoremnnedc 2254 Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
(DECID 𝐴 = 𝐵 → (¬ 𝐴𝐵𝐴 = 𝐵))

Theoremdcned 2255 Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
(𝜑DECID 𝐴 = 𝐵)       (𝜑DECID 𝐴𝐵)

Theoremneqned 2256 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2270. One-way deduction form of df-ne 2250. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2299. (Revised by Wolf Lammen, 22-Nov-2019.)
(𝜑 → ¬ 𝐴 = 𝐵)       (𝜑𝐴𝐵)

Theoremneqne 2257 From non equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴 = 𝐵𝐴𝐵)

Theoremneirr 2258 No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
¬ 𝐴𝐴

Theoremeqneqall 2259 A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
(𝐴 = 𝐵 → (𝐴𝐵𝜑))

Theoremdcne 2260 Decidable equality expressed in terms of . Basically the same as df-dc 777. (Contributed by Jim Kingdon, 14-Mar-2020.)
(DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵𝐴𝐵))

Theoremnonconne 2261 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
¬ (𝐴 = 𝐵𝐴𝐵)

Theoremneeq1 2262 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))

Theoremneeq2 2263 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Theoremneeq1i 2264 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)

Theoremneeq2i 2265 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)

Theoremneeq12i 2266 Inference for inequality. (Contributed by NM, 24-Jul-2012.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶𝐵𝐷)

Theoremneeq1d 2267 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))

Theoremneeq2d 2268 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))

Theoremneeq12d 2269 Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))

Theoremneneqd 2270 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)       (𝜑 → ¬ 𝐴 = 𝐵)

Theoremneneq 2271 From inequality to non equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴𝐵 → ¬ 𝐴 = 𝐵)

Theoremeqnetri 2272 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴 = 𝐵    &   𝐵𝐶       𝐴𝐶

Theoremeqnetrd 2273 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremeqnetrri 2274 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴 = 𝐵    &   𝐴𝐶       𝐵𝐶

Theoremeqnetrrd 2275 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐶)       (𝜑𝐵𝐶)

Theoremneeqtri 2276 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶

Theoremneeqtrd 2277 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)

Theoremneeqtrri 2278 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴𝐵    &   𝐶 = 𝐵       𝐴𝐶

Theoremneeqtrrd 2279 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)

Theoremsyl5eqner 2280 B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)
𝐵 = 𝐴    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theorem3netr3d 2281 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝐷)

Theorem3netr4d 2282 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝐷)

Theorem3netr3g 2283 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(𝜑𝐴𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝐷)

Theorem3netr4g 2284 Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝐷)

Theoremnecon3abii 2285 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
(𝐴 = 𝐵𝜑)       (𝐴𝐵 ↔ ¬ 𝜑)

Theoremnecon3bbii 2286 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
(𝜑𝐴 = 𝐵)       𝜑𝐴𝐵)

Theoremnecon3bii 2287 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
(𝐴 = 𝐵𝐶 = 𝐷)       (𝐴𝐵𝐶𝐷)

Theoremnecon3abid 2288 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
(𝜑 → (𝐴 = 𝐵𝜓))       (𝜑 → (𝐴𝐵 ↔ ¬ 𝜓))

Theoremnecon3bbid 2289 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
(𝜑 → (𝜓𝐴 = 𝐵))       (𝜑 → (¬ 𝜓𝐴𝐵))

Theoremnecon3bid 2290 Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))       (𝜑 → (𝐴𝐵𝐶𝐷))

Theoremnecon3ad 2291 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
(𝜑 → (𝜓𝐴 = 𝐵))       (𝜑 → (𝐴𝐵 → ¬ 𝜓))

Theoremnecon3bd 2292 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
(𝜑 → (𝐴 = 𝐵𝜓))       (𝜑 → (¬ 𝜓𝐴𝐵))

Theoremnecon3d 2293 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
(𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))       (𝜑 → (𝐶𝐷𝐴𝐵))

Theoremnesym 2294 Characterization of inequality in terms of reversed equality (see bicom 138). (Contributed by BJ, 7-Jul-2018.)
(𝐴𝐵 ↔ ¬ 𝐵 = 𝐴)

Theoremnesymi 2295 Inference associated with nesym 2294. (Contributed by BJ, 7-Jul-2018.)
𝐴𝐵        ¬ 𝐵 = 𝐴

Theoremnesymir 2296 Inference associated with nesym 2294. (Contributed by BJ, 7-Jul-2018.)
¬ 𝐴 = 𝐵       𝐵𝐴

Theoremnecon3i 2297 Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)
(𝐴 = 𝐵𝐶 = 𝐷)       (𝐶𝐷𝐴𝐵)

Theoremnecon3ai 2298 Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
(𝜑𝐴 = 𝐵)       (𝐴𝐵 → ¬ 𝜑)

Theoremnecon3bi 2299 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
(𝐴 = 𝐵𝜑)       𝜑𝐴𝐵)

Theoremnecon1aidc 2300 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
(DECID 𝜑 → (¬ 𝜑𝐴 = 𝐵))       (DECID 𝜑 → (𝐴𝐵𝜑))

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