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Theorem List for Metamath Proof Explorer - 2301-2400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem2sb6rf 2301* Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove variable constraints. (Revised by Wolf Lammen, 28-Sep-2018.)

Theoremsb7f 2302* This version of dfsb7 2304 does not require that and be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-5 1766 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1806 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsb7h 2303* This version of dfsb7 2304 does not require that and be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-5 1766 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1806 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremdfsb7 2304* An alternate definition of proper substitution df-sb 1806. By introducing a dummy variable in the definiens, we are able to eliminate any distinct variable restrictions among the variables , , and of the definiendum. No distinct variable conflicts arise because effectively insulates from . To achieve this, we use a chain of two substitutions in the form of sb5 2279, first for then for . Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2458. Theorem sb7h 2303 provides a version where and don't have to be distinct. (Contributed by NM, 28-Jan-2004.)

Theoremsb10f 2305* Hao Wang's identity axiom P6 in Irving Copi, Symbolic Logic (5th ed., 1979), p. 328. In traditional predicate calculus, this is a sole axiom for identity from which the usual ones can be derived. (Contributed by NM, 9-May-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsbid2v 2306* An identity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)

Theoremsbelx 2307* Elimination of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbel2x 2308* Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.)

Theoremsbal1 2309* A theorem used in elimination of disjoint variable restriction on and by replacing it with a distinctor . (Contributed by NM, 15-May-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2018.)

Theoremsbal2 2310* Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) Remove a distinct variable constraint. (Revised by Wolf Lammen, 3-Oct-2018.)

Theoremsbal 2311* Move universal quantifier in and out of substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.)

Theoremsbex 2312* Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)

Theoremsbalv 2313* Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)

Theoremsbco4lem 2314* Lemma for sbco4 2315. It replaces the temporary variable with another temporary variable . (Contributed by Jim Kingdon, 26-Sep-2018.)

Theoremsbco4 2315* Two ways of exchanging two variables. Both sides of the biconditional exchange and , either via two temporary variables and , or a single temporary . (Contributed by Jim Kingdon, 25-Sep-2018.)

Theorem2sb8e 2316* An equivalent expression for double existence. (Contributed by Wolf Lammen, 2-Nov-2019.)

Theoremexsb 2317* An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)

Theorem2exsb 2318* An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 30-Sep-2018.)

1.6  Existential uniqueness

Syntaxweu 2319 Extend wff definition to include existential uniqueness ("there exists a unique such that ").

Syntaxwmo 2320 Extend wff definition to include uniqueness ("there exists at most one such that ").

Theoremeujust 2321* A soundness justification theorem for df-eu 2323, showing that the definition is equivalent to itself with its dummy variable renamed. Note that and needn't be distinct variables. See eujustALT 2322 for a proof that provides an example of how it can be achieved through the use of dvelim 2186. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

TheoremeujustALT 2322* Alternate proof of eujust 2321 illustrating the use of dvelim 2186. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Definitiondf-eu 2323* 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 2359, eu2 2358, eu3v 2347, and eu5 2345 (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 2405). (Contributed by NM, 12-Aug-1993.)

Definitiondf-mo 2324 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 2356. For other possible definitions see mo2 2328 and mo4 2366. (Contributed by NM, 8-Mar-1995.)

Theoremeuequ1 2325* Equality has existential uniqueness. Special case of eueq1 3199 proved using only predicate calculus. The proof needs be free of . This is ensured by having and be distinct. Alternatively, a distinctor could have been used instead. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof shortened by Wolf Lammen, 8-Sep-2019.)

Theoremmo2v 2326* Alternate definition of "at most one." Unlike mo2 2328, which is slightly more general, it does not depend on ax-11 1937 and ax-13 2104, whence it is preferable within predicate logic. Elsewhere, most theorems depend on these axioms anyway, so this advantage is no longer important. (Contributed by Wolf Lammen, 27-May-2019.) (Proof shortened by Wolf Lammen, 10-Nov-2019.)

Theoremeuf 2327* A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2018.)

Theoremmo2 2328* Alternate definition of "at most one." (Contributed by NM, 8-Mar-1995.) Restrict dummy variable z. (Revised by Wolf Lammen, 28-May-2019.)

Theoremnfeu1 2329 Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfmo1 2330 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfeud2 2331 Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.)

Theoremnfmod2 2332 Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremnfeud 2333 Deduction version of nfeu 2335. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfmod 2334 Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremnfeu 2335 Bound-variable hypothesis builder for uniqueness. Note that and needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfmo 2336 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)

Theoremeubid 2337 Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Theoremmobid 2338 Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)

Theoremeubidv 2339* Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Theoremmobidv 2340* Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.)

Theoremeubii 2341 Introduce uniqueness quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremmobii 2342 Formula-building rule for "at most one" quantifier (inference rule). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)

Theoremeuex 2343 Existential uniqueness implies existence. For a shorter proof using more axioms, see euexALT 2360. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2018.)

Theoremexmo 2344 Something exists or at most one exists. (Contributed by NM, 8-Mar-1995.)

Theoremeu5 2345 Uniqueness in terms of "at most one." Revised to reduce dependencies on axioms. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 25-May-2019.)

Theoremexmoeu2 2346 Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)

Theoremeu3v 2347* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) Add a distinct variable condition on . (Revised by Wolf Lammen, 29-May-2019.)

Theoremeumo 2348 Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.)

Theoremeumoi 2349 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)

Theoremmoabs 2350 Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)

Theoremexmoeu 2351 Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.)

Theoremsb8eu 2352 Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.)

Theoremsb8mo 2353 Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremcbveu 2354 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremcbvmo 2355 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)

Theoremmo3 2356* 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.) (Proof shortened by Wolf Lammen, 18-Aug-2019.)

Theoremmo 2357* Equivalent definitions of "there exists at most one." (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Dec-2018.)

Theoremeu2 2358* An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) (Proof shortened by Wolf Lammen, 2-Dec-2018.)

Theoremeu1 2359* 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.) (Proof shortened by Wolf Lammen, 29-Oct-2018.)

TheoremeuexALT 2360 Alternate proof of euex 2343. Shorter but uses more axioms. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremeuor 2361 Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.)

Theoremeuorv 2362* Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremeuor2 2363 Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)

Theoremsbmo 2364* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremmo4f 2365* "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)

Theoremmo4 2366* "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Theoremeu4 2367* Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Theoremmoim 2368 "At most one" reverses implication. (Contributed by NM, 22-Apr-1995.)

Theoremmoimi 2369 "At most one" reverses implication. (Contributed by NM, 15-Feb-2006.)

Theoremmoa1 2370 If an implication holds for at most one value, then its consequent holds for at most one value. See also ala1 1718 and exa1 1719. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Wolf Lammen, 22-Dec-2018.) (Revised by BJ, 29-Mar-2021.)

Theoremeuimmo 2371 Uniqueness implies "at most one" through reverse implication. (Contributed by NM, 22-Apr-1995.)

Theoremeuim 2372 Add existential uniqueness quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremmoan 2373 "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)

Theoremmoani 2374 "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)

Theoremmoor 2375 "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)

Theoremmooran1 2376 "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmooran2 2377 "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmoanim 2378 Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)

Theoremeuan 2379 Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)

Theoremmoanimv 2380* Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremmoanmo 2381 Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.)

Theoremmoaneu 2382 Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)

Theoremeuanv 2383* Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)

Theoremmopick 2384 "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 17-Sep-2019.)

Theoremeupick 2385 Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing such that is true, and there is also an (actually the same one) such that and are both true, then implies regardless of . This theorem can be useful for eliminating existential quantifiers in a hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by NM, 10-Jul-1994.)

Theoremeupicka 2386 Version of eupick 2385 with closed formulas. (Contributed by NM, 6-Sep-2008.)

Theoremeupickb 2387 Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)

Theoremeupickbi 2388 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)

Theoremmopick2 2389 "At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1739. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremmoexex 2390 "At most one" double quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.)

Theoremmoexexv 2391* "At most one" double quantification. (Contributed by NM, 26-Jan-1997.)

Theorem2moex 2392 Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)

Theorem2euex 2393 Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theorem2eumo 2394 Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.)

Theorem2eu2ex 2395 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)

Theorem2moswap 2396 A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.)

Theorem2euswap 2397 A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.)

Theorem2exeu 2398 Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)

Theorem2mo2 2399* This theorem extends the idea of "at most one" to expressions in two set variables ("at most one pair and ". Note: this is not expressed by ). 2eu4 2405 relates this extension to double existential uniqueness, if at least one pair exists. (Contributed by Wolf Lammen, 26-Oct-2019.)

Theorem2mo 2400* Two equivalent expressions for double "at most one." (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Nov-2019.)

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