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

Theoremnelneq2 2201 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵 = 𝐶)

Theoremeqsb3lem 2202* Lemma for eqsb3 2203. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)

Theoremeqsb3 2203* Substitution applied to an atomic wff (class version of equsb3 1885). (Contributed by Rodolfo Medina, 28-Apr-2010.)
([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)

Theoremclelsb3 2204* Substitution applied to an atomic wff (class version of elsb3 1912). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)

Theoremclelsb4 2205* Substitution applied to an atomic wff (class version of elsb4 1913). (Contributed by Jim Kingdon, 22-Nov-2018.)
([𝑥 / 𝑦]𝐴𝑦𝐴𝑥)

Theoremhbxfreq 2206 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1416 for equivalence version. (Contributed by NM, 21-Aug-2007.)
𝐴 = 𝐵    &   (𝑦𝐵 → ∀𝑥 𝑦𝐵)       (𝑦𝐴 → ∀𝑥 𝑦𝐴)

Theoremhblem 2207* 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 2208* 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 2213 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 2209* Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))

Theoremabeq2i 2210 Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 3-Apr-1996.)
𝐴 = {𝑥𝜑}       (𝑥𝐴𝜑)

Theoremabeq1i 2211 Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 31-Jul-1994.)
{𝑥𝜑} = 𝐴       (𝜑𝑥𝐴)

Theoremabeq2d 2212 Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
(𝜑𝐴 = {𝑥𝜓})       (𝜑 → (𝑥𝐴𝜓))

Theoremabbi 2213 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 2214* Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 5-Aug-1993.)
(𝑥𝐴𝜑)       𝐴 = {𝑥𝜑}

Theoremabbii 2215 Equivalent wff's yield equal class abstractions (inference form). (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       {𝑥𝜑} = {𝑥𝜓}

Theoremabbid 2216 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 2217* Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 10-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} = {𝑥𝜒})

Theoremabbi2dv 2218* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
(𝜑 → (𝑥𝐴𝜓))       (𝜑𝐴 = {𝑥𝜓})

Theoremabbi1dv 2219* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
(𝜑 → (𝜓𝑥𝐴))       (𝜑 → {𝑥𝜓} = 𝐴)

Theoremabid2 2220* A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)
{𝑥𝑥𝐴} = 𝐴

Theoremsb8ab 2221 Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)
𝑦𝜑       {𝑥𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑}

Theoremcbvab 2222 Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}

Theoremcbvabv 2223* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)
(𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}

Theoremclelab 2224* Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
(𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑))

Theoremclabel 2225* Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
({𝑥𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦𝐴 ∧ ∀𝑥(𝑥𝑦𝜑)))

Theoremsbab 2226* 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 2227 Extend wff definition to include the not-free predicate for classes.
wff 𝑥𝐴

Theoremnfcjust 2228* Justification theorem for df-nfc 2229. (Contributed by Mario Carneiro, 13-Oct-2016.)
(∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)

Definitiondf-nfc 2229* 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 1405 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)

Theoremnfci 2230* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥 𝑦𝐴       𝑥𝐴

Theoremnfcii 2231* Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       𝑥𝐴

Theoremnfcr 2232* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑥𝐴 → Ⅎ𝑥 𝑦𝐴)

Theoremnfcrii 2233* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴       (𝑦𝐴 → ∀𝑥 𝑦𝐴)

Theoremnfcri 2234* Consequence of the not-free predicate. (Note that unlike nfcr 2232, this does not require 𝑦 and 𝐴 to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴       𝑥 𝑦𝐴

Theoremnfcd 2235* 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 2236 Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝐴 = 𝐵       (𝑥𝐴𝑥𝐵)

Theoremnfcxfr 2237 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 2238 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 2239 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴𝑥𝐵))

Theoremnfcv 2240* If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴

Theoremnfcvd 2241* If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)

Theoremnfab1 2242 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥{𝑥𝜑}

Theoremnfnfc1 2243 𝑥 is bound in 𝑥𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝑥𝐴

Theoremclelsb3f 2244 Substitution applied to an atomic wff (class version of elsb3 1912). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
𝑦𝐴       ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)

Theoremnfab 2245 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       𝑥{𝑦𝜑}

Theoremnfaba1 2246 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑥{𝑦 ∣ ∀𝑥𝜑}

Theoremnfnfc 2247 Hypothesis builder for 𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴       𝑥𝑦𝐴

Theoremnfeq 2248 Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴 = 𝐵

Theoremnfel 2249 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵

Theoremnfeq1 2250* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴       𝑥 𝐴 = 𝐵

Theoremnfel1 2251* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴       𝑥 𝐴𝐵

Theoremnfeq2 2252* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐵       𝑥 𝐴 = 𝐵

Theoremnfel2 2253* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
𝑥𝐵       𝑥 𝐴𝐵

Theoremnfcrd 2254* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑𝑥𝐴)       (𝜑 → Ⅎ𝑥 𝑦𝐴)

Theoremnfeqd 2255 Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)

Theoremnfeld 2256 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝐵)

Theoremdrnfc1 2257 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 2258 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 2259 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥{𝑦𝜓})

Theoremdvelimdc 2260 Deduction form of dvelimc 2261. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑥𝜑    &   𝑧𝜑    &   (𝜑𝑥𝐴)    &   (𝜑𝑧𝐵)    &   (𝜑 → (𝑧 = 𝑦𝐴 = 𝐵))       (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵))

Theoremdvelimc 2261 Version of dvelim 1953 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑥𝐴    &   𝑧𝐵    &   (𝑧 = 𝑦𝐴 = 𝐵)       (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵)

Theoremnfcvf 2262 If 𝑥 and 𝑦 are distinct, then 𝑥 is not free in 𝑦. (Contributed by Mario Carneiro, 8-Oct-2016.)
(¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)

Theoremnfcvf2 2263 If 𝑥 and 𝑦 are distinct, then 𝑦 is not free in 𝑥. (Contributed by Mario Carneiro, 5-Dec-2016.)
(¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)

Theoremcleqf 2264 Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2199. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Theoremabid2f 2265 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 2266* 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 2267 Extend wff notation to include inequality.
wff 𝐴𝐵

Definitiondf-ne 2268 Define inequality. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)

Theoremneii 2269 Inference associated with df-ne 2268. (Contributed by BJ, 7-Jul-2018.)
𝐴𝐵        ¬ 𝐴 = 𝐵

Theoremneir 2270 Inference associated with df-ne 2268. (Contributed by BJ, 7-Jul-2018.)
¬ 𝐴 = 𝐵       𝐴𝐵

Theoremnner 2271 Negation of inequality. (Contributed by Jim Kingdon, 23-Dec-2018.)
(𝐴 = 𝐵 → ¬ 𝐴𝐵)

Theoremnnedc 2272 Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
(DECID 𝐴 = 𝐵 → (¬ 𝐴𝐵𝐴 = 𝐵))

Theoremdcned 2273 Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
(𝜑DECID 𝐴 = 𝐵)       (𝜑DECID 𝐴𝐵)

Theoremneqned 2274 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2288. One-way deduction form of df-ne 2268. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2317. (Revised by Wolf Lammen, 22-Nov-2019.)
(𝜑 → ¬ 𝐴 = 𝐵)       (𝜑𝐴𝐵)

Theoremneqne 2275 From non-equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴 = 𝐵𝐴𝐵)

Theoremneirr 2276 No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
¬ 𝐴𝐴

Theoremeqneqall 2277 A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
(𝐴 = 𝐵 → (𝐴𝐵𝜑))

Theoremdcne 2278 Decidable equality expressed in terms of . Basically the same as df-dc 787. (Contributed by Jim Kingdon, 14-Mar-2020.)
(DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵𝐴𝐵))

Theoremnonconne 2279 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
¬ (𝐴 = 𝐵𝐴𝐵)

Theoremneeq1 2280 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))

Theoremneeq2 2281 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Theoremneeq1i 2282 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)

Theoremneeq2i 2283 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)

Theoremneeq12i 2284 Inference for inequality. (Contributed by NM, 24-Jul-2012.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶𝐵𝐷)

Theoremneeq1d 2285 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))

Theoremneeq2d 2286 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))

Theoremneeq12d 2287 Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))

Theoremneneqd 2288 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)       (𝜑 → ¬ 𝐴 = 𝐵)

Theoremneneq 2289 From inequality to non-equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴𝐵 → ¬ 𝐴 = 𝐵)

Theoremeqnetri 2290 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴 = 𝐵    &   𝐵𝐶       𝐴𝐶

Theoremeqnetrd 2291 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremeqnetrri 2292 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴 = 𝐵    &   𝐴𝐶       𝐵𝐶

Theoremeqnetrrd 2293 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐶)       (𝜑𝐵𝐶)

Theoremneeqtri 2294 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶

Theoremneeqtrd 2295 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)

Theoremneeqtrri 2296 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
𝐴𝐵    &   𝐶 = 𝐵       𝐴𝐶

Theoremneeqtrrd 2297 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)

Theoremsyl5eqner 2298 B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)
𝐵 = 𝐴    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theorem3netr3d 2299 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝐷)

Theorem3netr4d 2300 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝐷)

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