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

Theoremhbae-o 2201 All variables are effectively bound in an identical variable specifier. Version of hbae 2005 using ax-10o 2187. (Contributed by NM, 5-Aug-1993.) (Proof modification is disccouraged.) (New usage is discouraged.)

Theoremdral1-o 2202 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral1 2022 using ax-10o 2187. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)

Theoremax11 2203 Rederivation of axiom ax-11 1757 from ax-11o 2189, ax-10o 2187, and other older axioms. The proof does not require ax-16 2192 or ax-17 1623. See theorem ax11o 2045 for the derivation of ax-11o 2189 from ax-11 1757.

An open problem is whether we can prove this using ax-10 2188 instead of ax-10o 2187.

This proof uses newer axioms ax-5 1563 and ax-9 1662, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-5o 2184 and ax-9o 2186. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax12from12o 2204 Derive ax-12 1946 from ax-12o 2190 and other older axioms.

This proof uses newer axioms ax-5 1563 and ax-9 1662, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-5o 2184 and ax-9o 2186. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

1.6.3  Legacy theorems using obsolete axioms

These theorems were mostly intended to study properties of the older axiom schemes and are not useful outside of this section. They should not be used outside of this section. They may be deleted when they are deemed to no longer be of interest.

Theoremax17o 2205* Axiom to quantify a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113.

(This theorem simply repeats ax-17 1623 so that we can include the following note, which applies only to the obsolete axiomatization.)

This axiom is logically redundant in the (logically complete) predicate calculus axiom system consisting of ax-gen 1552, ax-5o 2184, ax-4 2183, ax-7 1745, ax-6o 2185, ax-8 1683, ax-12o 2190, ax-9o 2186, ax-10o 2187, ax-13 1723, ax-14 1725, ax-15 2191, ax-11o 2189, and ax-16 2192: in that system, we can derive any instance of ax-17 1623 not containing wff variables by induction on formula length, using ax17eq 2231 and ax17el 2237 for the basis together hbn 1797, hbal 1747, and hbim 1832. However, if we omit this axiom, our development would be quite inconvenient since we could work only with specific instances of wffs containing no wff variables - this axiom introduces the concept of a set variable not occurring in a wff (as opposed to just two set variables being distinct). (Contributed by NM, 19-Aug-2017.) (New usage is discouraged.) (Proof modification discouraged.)

Theoremequid1 2206 Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms (although the proof, as you can see below, is not as obvious as you might think). This proof uses only axioms without distinct variable conditions and thus requires no dummy variables. A simpler proof, similar to Tarki's, is possible if we make use of ax-17 1623; see the proof of equid 1684. See equid1ALT 2224 for an alternate proof. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsps-o 2207 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremhbequid 2208 Bound-variable hypothesis builder for . This theorem tells us that any variable, including , is effectively not free in , even though is technically free according to the traditional definition of free variable. (The proof does not use ax-9o 2186.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnfequid-o 2209 Bound-variable hypothesis builder for . This theorem tells us that any variable, including , is effectively not free in , even though is technically free according to the traditional definition of free variable. (The proof uses only ax-5 1563, ax-8 1683, ax-12o 2190, and ax-gen 1552. This shows that this can be proved without ax9 1949, even though the theorem equid 1684 cannot be. A shorter proof using ax9 1949 is obtainable from equid 1684 and hbth 1558.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax9v 1663, which is used for the derivation of ax12o 1976, unless we consider ax-12o 2190 the starting axiom rather than ax-12 1946. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax46 2210 Proof of a single axiom that can replace ax-4 2183 and ax-6o 2185. See ax46to4 2211 and ax46to6 2212 for the re-derivation of those axioms. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax46to4 2211 Re-derivation of ax-4 2183 from ax46 2210. Only propositional calculus is used for the re-derivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax46to6 2212 Re-derivation of ax-6o 2185 from ax46 2210. Only propositional calculus is used for the re-derivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax67 2213 Proof of a single axiom that can replace both ax-6o 2185 and ax-7 1745. See ax67to6 2215 and ax67to7 2216 for the re-derivation of those axioms. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnfa1-o 2214 is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax67to6 2215 Re-derivation of ax-6o 2185 from ax67 2213. Note that ax-6o 2185 and ax-7 1745 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax67to7 2216 Re-derivation of ax-7 1745 from ax67 2213. Note that ax-6o 2185 and ax-7 1745 are not used by the re-derivation. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax467 2217 Proof of a single axiom that can replace ax-4 2183, ax-6o 2185, and ax-7 1745 in a subsystem that includes these axioms plus ax-5o 2184 and ax-gen 1552 (and propositional calculus). See ax467to4 2218, ax467to6 2219, and ax467to7 2220 for the re-derivation of those axioms. This theorem extends the idea in Scott Fenton's ax46 2210. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax467to4 2218 Re-derivation of ax-4 2183 from ax467 2217. Only propositional calculus is used by the re-derivation. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax467to6 2219 Re-derivation of ax-6o 2185 from ax467 2217. Note that ax-6o 2185 and ax-7 1745 are not used by the re-derivation. The use of alimi 1565 (which uses ax-4 2183) is allowed since we have already proved ax467to4 2218. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax467to7 2220 Re-derivation of ax-7 1745 from ax467 2217. Note that ax-6o 2185 and ax-7 1745 are not used by the re-derivation. The use of alimi 1565 (which uses ax-4 2183) is allowed since we have already proved ax467to4 2218. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremequidqe 2221 equid 1684 with existential quantifier without using ax-4 2183 or ax-17 1623. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) (Proof modification is discouraged.)

Theoremax4sp1 2222 A special case of ax-4 2183 without using ax-4 2183 or ax-17 1623. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.)

Theoremequidq 2223 equid 1684 with universal quantifier without using ax-4 2183 or ax-17 1623. (Contributed by NM, 13-Jan-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremequid1ALT 2224 Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. Alternate proof of equid1 2206 from older axioms ax-6o 2185 and ax-9o 2186. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax10from10o 2225 Rederivation of ax-10 2188 from original version ax-10o 2187. See theorem ax10o 2001 for the derivation of ax-10o 2187 from ax-10 2188.

This theorem should not be referenced in any proof. Instead, use ax-10 2188 above so that uses of ax-10 2188 can be more easily identified, or use aecom-o 2199 when this form is needed for studies involving ax-10o 2187 and omitting ax-17 1623. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnaecoms-o 2226 A commutation rule for distinct variable specifiers. Version of naecoms 2004 using ax-10o 2187. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremhbnae-o 2227 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Version of hbnae 2007 using ax-10o 2187. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdvelimf-o 2228 Proof of dvelimh 2015 that uses ax-10o 2187 but not ax-11o 2189, ax-10 2188, or ax-11 1757. Version of dvelimh 2015 using ax-10o 2187 instead of ax10o 2001. (Contributed by NM, 12-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdral2-o 2229 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral2 2020 using ax-10o 2187. (Contributed by NM, 27-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremaev-o 2230* A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 2192. Version of aev 2011 using ax-10o 2187. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax17eq 2231* Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 1623 considered as a metatheorem. Do not use it for later proofs - use ax-17 1623 instead, to avoid reference to the redundant axiom ax-16 2192.) (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdveeq2-o 2232* Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2019 using ax-11o 2189. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdveeq2-o16 2233* Version of dveeq2 2019 using ax-16 2192 instead of ax-17 1623. TO DO: Recover proof from older set.mm to remove use of ax-17 1623. (Contributed by NM, 29-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorema16g-o 2234* A generalization of axiom ax-16 2192. Version of a16g 2012 using ax-10o 2187. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdveeq1-o 2235* Quantifier introduction when one pair of variables is distinct. Version of dveeq1 1987 using ax-10o . (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdveeq1-o16 2236* Version of dveeq1 1987 using ax-16 2192 instead of ax-17 1623. (Contributed by NM, 29-Apr-2008.) TO DO: Recover proof from older set.mm to remove use of ax-17 1623. (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax17el 2237* Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-17 1623 considered as a metatheorem.) (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax10-16 2238* This theorem shows that, given ax-16 2192, we can derive a version of ax-10 2188. However, it is weaker than ax-10 2188 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdveel2ALT 2239* Version of dveel2 2067 using ax-16 2192 instead of ax-17 1623. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11f 2240 Basis step for constructing a substitution instance of ax-11o 2189 without using ax-11o 2189. We can start with any formula in which is not free. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11eq 2241 Basis step for constructing a substitution instance of ax-11o 2189 without using ax-11o 2189. Atomic formula for equality predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11el 2242 Basis step for constructing a substitution instance of ax-11o 2189 without using ax-11o 2189. Atomic formula for membership predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11indn 2243 Induction step for constructing a substitution instance of ax-11o 2189 without using ax-11o 2189. Negation case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11indi 2244 Induction step for constructing a substitution instance of ax-11o 2189 without using ax-11o 2189. Implication case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11indalem 2245 Lemma for ax11inda2 2247 and ax11inda 2248. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11inda2ALT 2246* A proof of ax11inda2 2247 that is slightly more direct. (Contributed by NM, 4-May-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11inda2 2247* Induction step for constructing a substitution instance of ax-11o 2189 without using ax-11o 2189. Quantification case. When and are distinct, this theorem avoids the dummy variables needed by the more general ax11inda 2248. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11inda 2248* Induction step for constructing a substitution instance of ax-11o 2189 without using ax-11o 2189. Quantification case. (When and are distinct, ax11inda2 2247 may be used instead to avoid the dummy variable in the proof.) (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11v2-o 2249* Recovery of ax-11o 2189 from ax11v 2143 without using ax-11o 2189. The hypothesis is even weaker than ax11v 2143, with both distinct from and not occurring in . Thus, the hypothesis provides an alternate axiom that can be used in place of ax-11o 2189. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11a2-o 2250* Derive ax-11o 2189 from a hypothesis in the form of ax-11 1757, without using ax-11 1757 or ax-11o 2189. The hypothesis is even weaker than ax-11 1757, with both distinct from and not occurring in . Thus, the hypothesis provides an alternate axiom that can be used in place of ax-11 1757, if we also hvae ax-10o 2187 which this proof uses . As theorem ax11 2203 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-10 2188 instead of ax-10o 2187. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax10o-o 2251 Show that ax-10o 2187 can be derived from ax-10 2188. An open problem is whether this theorem can be derived from ax-10 2188 and the others when ax-11 1757 is replaced with ax-11o 2189. See theorem ax10from10o 2225 for the rederivation of ax-10 2188 from ax10o 2001.

Normally, ax10o 2001 should be used rather than ax-10o 2187 or ax10o-o 2251, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

1.7  Existential uniqueness

Syntaxweu 2252 Extend wff definition to include existential uniqueness ("there exists a unique such that ").

Syntaxwmo 2253 Extend wff definition to include uniqueness ("there exists at most one such that ").

Theoremeujust 2254* A soundness justification theorem for df-eu 2256, showing that the definition is equivalent to itself with its dummy variable renamed. Note that and needn't be distinct variables. See eujustALT 2255 for a proof that provides an example of how it can be achieved through the use of dvelim 2064. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

TheoremeujustALT 2255* A soundness justification theorem for df-eu 2256, showing that the definition is equivalent to itself with its dummy variable renamed. Note that and needn't be distinct variables. While this isn't strictly necessary for soundness, the proof provides an example of how it can be achieved through the use of dvelim 2064. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Definitiondf-eu 2256* Define existential uniqueness, i.e. "there exists exactly one such that ." Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 2273, eu2 2277, eu3 2278, and eu5 2290 (which in some cases we show with a hypothesis in place of a distinct variable condition on and ). Double uniqueness is tricky: does not mean "exactly one and one " (see 2eu4 2335). (Contributed by NM, 12-Aug-1993.)

Definitiondf-mo 2257 Define "there exists at most one such that ." Here we define it in terms of existential uniqueness. Notation of [BellMachover] p. 460, whose definition we show as mo3 2283. For other possible definitions see mo2 2281 and mo4 2285. (Contributed by NM, 8-Mar-1995.)

Theoremeuf 2258* A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)

Theoremeubid 2259 Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Theoremeubidv 2260* Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Theoremeubii 2261 Introduce uniqueness quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremnfeu1 2262 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfmo1 2263 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfeud2 2264 Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremnfmod2 2265 Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremnfeud 2266 Deduction version of nfeu 2268. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfmod 2267 Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremnfeu 2268 Bound-variable hypothesis builder for "at most one." Note that and needn't be distinct (this makes the proof more difficult). (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfmo 2269 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)

Theoremsb8eu 2270 Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremsb8mo 2271 Variable substitution in uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremcbveu 2272 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremeu1 2273* An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremmo 2274* Equivalent definitions of "there exists at most one." (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremeuex 2275 Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremeumo0 2276* Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)

Theoremeu2 2277* An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.)

Theoremeu3 2278* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)

Theoremeuor 2279 Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.)

Theoremeuorv 2280* Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremmo2 2281* Alternate definition of "at most one." (Contributed by NM, 8-Mar-1995.)

Theoremsbmo 2282* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremmo3 2283* Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that not occur in in place of our hypothesis. (Contributed by NM, 8-Mar-1995.)

Theoremmo4f 2284* "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)

Theoremmo4 2285* "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Theoremmobid 2286 Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)

Theoremmobidv 2287* Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.)

Theoremmobii 2288 Formula-building rule for "at most one" quantifier (inference rule). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)

Theoremcbvmo 2289 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)

Theoremeu5 2290 Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.)

Theoremeu4 2291* Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Theoremeumo 2292 Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.)

Theoremeumoi 2293 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)

Theoremexmoeu 2294 Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.)

Theoremexmoeu2 2295 Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)

Theoremmoabs 2296 Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)

Theoremexmo 2297 Something exists or at most one exists. (Contributed by NM, 8-Mar-1995.)

Theoremmoim 2298 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)

Theoremmoimi 2299 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.)

Theoremmorimv 2300* Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)

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