P. 24 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
1.6  Existential uniqueness
Syntax	weu 2301	Extend wff definition to include existential uniqueness ("there exists a unique x such that ph").
wff E! x ph
Syntax	wmo 2302	Extend wff definition to include uniqueness ("there exists at most one x such that ph").
wff E* x ph
Theorem	eujust 2303*	A soundness justification theorem for df-eu 2305, showing that the definition is equivalent to itself with its dummy variable renamed. Note that y and z needn't be distinct variables. See eujustALT 2304 for a proof that provides an example of how it can be achieved through the use of dvelim 2173. (Contributed by NM, 11-Mar-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( E. y A. x ( ph <-> x = y ) <-> E. z A. x ( ph <-> x = z ) )
Theorem	eujustALT 2304*	Alternate proof of eujust 2303 illustrating the use of dvelim 2173. (Contributed by NM, 11-Mar-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( E. y A. x ( ph <-> x = y ) <-> E. z A. x ( ph <-> x = z ) )
Definition	df-eu 2305*	Define existential uniqueness, i.e. "there exists exactly one x such that ph." Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 2341, eu2 2340, eu3v 2329, and eu5 2327 (which in some cases we show with a hypothesis ph -> A. y ph in place of a distinct variable condition on y and ph). Double uniqueness is tricky: E! x E! y ph does not mean "exactly one x and one y " (see 2eu4 2387). (Contributed by NM, 12-Aug-1993.)
|- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
Definition	df-mo 2306	Define "there exists at most one x such that ph." Here we define it in terms of existential uniqueness. Notation of [BellMachover] p. 460, whose definition we show as mo3 2338. For other possible definitions see mo2 2310 and mo4 2348. (Contributed by NM, 8-Mar-1995.)
|- ( E* x ph <-> ( E. x ph -> E! x ph ) )
Theorem	euequ1 2307*	Equality has existential uniqueness. Special case of eueq1 3213 proved using only predicate calculus. The proof needs y = z be free of x. This is ensured by having x and y be distinct. Alternatively, a distinctor -. A. x x = y could have been used instead. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof shortened by Wolf Lammen, 8-Sep-2019.)
|- E! x x = y
Theorem	mo2v 2308*	Alternate definition of "at most one." Unlike mo2 2310, which is slightly more general, it does not depend on ax-11 1922 and ax-13 2093, 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.)
|- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
Theorem	euf 2309*	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.)
|- F/ y ph   =>    |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
Theorem	mo2 2310*	Alternate definition of "at most one." (Contributed by NM, 8-Mar-1995.) Restrict dummy variable z. (Revised by Wolf Lammen, 28-May-2019.)
|- F/ y ph   =>    |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
Theorem	nfeu1 2311	Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/ x E! x ph
Theorem	nfmo1 2312	Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/ x E* x ph
Theorem	nfeud2 2313	Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.)
|- F/ y ph   &    |- ( ( ph /\ -. A. x x = y ) -> F/ x ps )   =>    |- ( ph -> F/ x E! y ps )
Theorem	nfmod2 2314	Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)
|- F/ y ph   &    |- ( ( ph /\ -. A. x x = y ) -> F/ x ps )   =>    |- ( ph -> F/ x E* y ps )
Theorem	nfeud 2315	Deduction version of nfeu 2317. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x E! y ps )
Theorem	nfmod 2316	Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x E* y ps )
Theorem	nfeu 2317	Bound-variable hypothesis builder for uniqueness. Note that x and y needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/ x ph   =>    |- F/ x E! y ph
Theorem	nfmo 2318	Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)
|- F/ x ph   =>    |- F/ x E* y ph
Theorem	eubid 2319	Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)
|- F/ x ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E! x ps <-> E! x ch ) )
Theorem	mobid 2320	Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)
|- F/ x ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E* x ps <-> E* x ch ) )
Theorem	eubidv 2321*	Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E! x ps <-> E! x ch ) )
Theorem	mobidv 2322*	Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E* x ps <-> E* x ch ) )
Theorem	eubii 2323	Introduce uniqueness quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
|- ( ph <-> ps )   =>    |- ( E! x ph <-> E! x ps )
Theorem	mobii 2324	Formula-building rule for "at most one" quantifier (inference rule). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)
|- ( ps <-> ch )   =>    |- ( E* x ps <-> E* x ch )
Theorem	euex 2325	Existential uniqueness implies existence. For a shorter proof using more axioms, see euexALT 2342. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2018.)
|- ( E! x ph -> E. x ph )
Theorem	exmo 2326	Something exists or at most one exists. (Contributed by NM, 8-Mar-1995.)
|- ( E. x ph \/ E* x ph )
Theorem	eu5 2327	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.)
|- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
Theorem	exmoeu2 2328	Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)
|- ( E. x ph -> ( E* x ph <-> E! x ph ) )
Theorem	eu3v 2329*	An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) Add a distinct variable condition on ph. (Revised by Wolf Lammen, 29-May-2019.)
|- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
Theorem	eumo 2330	Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.)
|- ( E! x ph -> E* x ph )
Theorem	eumoi 2331	"At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
|- E! x ph   =>    |- E* x ph
Theorem	moabs 2332	Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)
|- ( E* x ph <-> ( E. x ph -> E* x ph ) )
Theorem	exmoeu 2333	Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.)
|- ( E. x ph <-> ( E* x ph -> E! x ph ) )
Theorem	sb8eu 2334	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.)
|- F/ y ph   =>    |- ( E! x ph <-> E! y [ y / x ] ph )
Theorem	sb8mo 2335	Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
|- F/ y ph   =>    |- ( E* x ph <-> E* y [ y / x ] ph )
Theorem	cbveu 2336	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x ph <-> E! y ps )
Theorem	cbvmo 2337	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E* x ph <-> E* y ps )
Theorem	mo3 2338*	Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that y not occur in ph in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Aug-2019.)
|- F/ y ph   =>    |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
Theorem	mo 2339*	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.)
|- F/ y ph   =>    |- ( E. y A. x ( ph -> x = y ) <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
Theorem	eu2 2340*	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.)
|- F/ y ph   =>    |- ( E! x ph <-> ( E. x ph /\ A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )
Theorem	eu1 2341*	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.)
|- F/ y ph   =>    |- ( E! x ph <-> E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) )
Theorem	euexALT 2342	Alternate proof of euex 2325. 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.)
|- ( E! x ph -> E. x ph )
Theorem	euor 2343	Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.)
|- F/ x ph   =>    |- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) )
Theorem	euorv 2344*	Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
|- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) )
Theorem	euor2 2345	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.)
|- ( -. E. x ph -> ( E! x ( ph \/ ps ) <-> E! x ps ) )
Theorem	sbmo 2346*	Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( [ y / x ] E* z ph <-> E* z [ y / x ] ph )
Theorem	mo4f 2347*	"At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)
|- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )
Theorem	mo4 2348*	"At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )
Theorem	eu4 2349*	Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x ph <-> ( E. x ph /\ A. x A. y ( ( ph /\ ps ) -> x = y ) ) )
Theorem	moim 2350	"At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)
|- ( A. x ( ph -> ps ) -> ( E* x ps -> E* x ph ) )
Theorem	moimi 2351	"At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.)
|- ( ph -> ps )   =>    |- ( E* x ps -> E* x ph )
Theorem	morimOLD 2352	TODO-NM: AV and BJ propose to keep this theorem as morim . Obsolete theorem as of 22-Dec-2018. Use moimi 2351 applied to ax-1 6, as demonstrated in the proof. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Wolf Lammen, 22-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( E* x ( ph -> ps ) -> ( ph -> E* x ps ) )
Theorem	euimmo 2353	Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.)
|- ( A. x ( ph -> ps ) -> ( E! x ps -> E* x ph ) )
Theorem	euim 2354	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.)
|- ( ( E. x ph /\ A. x ( ph -> ps ) ) -> ( E! x ps -> E! x ph ) )
Theorem	moan 2355	"At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)
|- ( E* x ph -> E* x ( ps /\ ph ) )
Theorem	moani 2356	"At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)
|- E* x ph   =>    |- E* x ( ps /\ ph )
Theorem	moor 2357	"At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)
|- ( E* x ( ph \/ ps ) -> E* x ph )
Theorem	mooran1 2358	"At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( ( E* x ph \/ E* x ps ) -> E* x ( ph /\ ps ) )
Theorem	mooran2 2359	"At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( E* x ( ph \/ ps ) -> ( E* x ph /\ E* x ps ) )
Theorem	moanim 2360	Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
|- F/ x ph   =>    |- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) )
Theorem	euan 2361	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.)
|- F/ x ph   =>    |- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) )
Theorem	moanimv 2362*	Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.)
|- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) )
Theorem	moanmo 2363	Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.)
|- E* x ( ph /\ E* x ph )
Theorem	moaneu 2364	Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
|- E* x ( ph /\ E! x ph )
Theorem	euanv 2365*	Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
|- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) )
Theorem	mopick 2366	"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.)
|- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
Theorem	eupick 2367	Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing x such that ph is true, and there is also an x (actually the same one) such that ph and ps are both true, then ph implies ps regardless of x. 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.)
|- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
Theorem	eupicka 2368	Version of eupick 2367 with closed formulas. (Contributed by NM, 6-Sep-2008.)
|- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> A. x ( ph -> ps ) )
Theorem	eupickb 2369	Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
|- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) -> ( ph <-> ps ) )
Theorem	eupickbi 2370	Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
|- ( E! x ph -> ( E. x ( ph /\ ps ) <-> A. x ( ph -> ps ) ) )
Theorem	mopick2 2371	"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 1733. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( ( E* x ph /\ E. x ( ph /\ ps ) /\ E. x ( ph /\ ch ) ) -> E. x ( ph /\ ps /\ ch ) )
Theorem	moexex 2372	"At most one" double quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.)
|- F/ y ph   =>    |- ( ( E* x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) )
Theorem	moexexv 2373*	"At most one" double quantification. (Contributed by NM, 26-Jan-1997.)
|- ( ( E* x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) )
Theorem	2moex 2374	Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)
|- ( E* x E. y ph -> A. y E* x ph )
Theorem	2euex 2375	Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( E! x E. y ph -> E. y E! x ph )
Theorem	2eumo 2376	Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.)
|- ( E! x E* y ph -> E* x E! y ph )
Theorem	2eu2ex 2377	Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
|- ( E! x E! y ph -> E. x E. y ph )
Theorem	2moswap 2378	A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
|- ( A. x E* y ph -> ( E* x E. y ph -> E* y E. x ph ) )
Theorem	2euswap 2379	A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
|- ( A. x E* y ph -> ( E! x E. y ph -> E! y E. x ph ) )
Theorem	2exeu 2380	Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)
|- ( ( E! x E. y ph /\ E! y E. x ph ) -> E! x E! y ph )
Theorem	2mo2 2381*	This theorem extends the idea of "at most one" to expressions in two set variables ("at most one pair x and y". Note: this is not expressed by E* x E* y ph). 2eu4 2387 relates this extension to double existential uniqueness, if at least one pair exists. (Contributed by Wolf Lammen, 26-Oct-2019.)
|- ( ( E* x E. y ph /\ E* y E. x ph ) <-> E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) )
Theorem	2mo 2382*	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.)
|- ( E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) <-> A. x A. y A. z A. w ( ( ph /\ [ z / x ] [ w / y ] ph ) -> ( x = z /\ y = w ) ) )
Theorem	2mos 2383*	Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.)
|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )   =>    |- ( E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) <-> A. x A. y A. z A. w ( ( ph /\ ps ) -> ( x = z /\ y = w ) ) )
Theorem	2eu1 2384	Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 11-Nov-2019.)
|- ( A. x E* y ph -> ( E! x E! y ph <-> ( E! x E. y ph /\ E! y E. x ph ) ) )
Theorem	2eu2 2385	Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
|- ( E! y E. x ph -> ( E! x E! y ph <-> E! x E. y ph ) )
Theorem	2eu3 2386	Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
|- ( A. x A. y ( E* x ph \/ E* y ph ) -> ( ( E! x E! y ph /\ E! y E! x ph ) <-> ( E! x E. y ph /\ E! y E. x ph ) ) )
Theorem	2eu4 2387*	This theorem provides us with a definition of double existential uniqueness ("exactly one x and exactly one y"). Naively one might think (incorrectly) that it could be defined by E! x E! y ph. See 2eu1 2384 for a condition under which the naive definition holds and 2exeu 2380 for a one-way implication. See 2eu5 2388 and 2eu8 2391 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.)
|- ( ( E! x E. y ph /\ E! y E. x ph ) <-> ( E. x E. y ph /\ E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) ) )
Theorem	2eu5 2388*	An alternate definition of double existential uniqueness (see 2eu4 2387). A mistake sometimes made in the literature is to use E! x E! y to mean "exactly one x and exactly one y." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining A. x E* y ph as an additional condition. The correct definition apparently has never been published. ( E* means "exists at most one."). (Contributed by NM, 26-Oct-2003.)
|- ( ( E! x E! y ph /\ A. x E* y ph ) <-> ( E. x E. y ph /\ E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) ) )
Theorem	2eu6 2389*	Two equivalent expressions for double existential uniqueness. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Oct-2019.)
|- ( ( E! x E. y ph /\ E! y E. x ph ) <-> E. z E. w A. x A. y ( ph <-> ( x = z /\ y = w ) ) )
Theorem	2eu7 2390	Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)
|- ( ( E! x E. y ph /\ E! y E. x ph ) <-> E! x E! y ( E. x ph /\ E. y ph ) )
Theorem	2eu8 2391	Two equivalent expressions for double existential uniqueness. Curiously, we can put E! on either of the internal conjuncts but not both. We can also commute E! x E! y using 2eu7 2390. (Contributed by NM, 20-Feb-2005.)
|- ( E! x E! y ( E. x ph /\ E. y ph ) <-> E! x E! y ( E! x ph /\ E. y ph ) )
Theorem	exists1 2392*	Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory; see theorem dtru 4597. (Contributed by NM, 5-Apr-2004.)
|- ( E! x x = x <-> A. x x = y )
Theorem	exists2 2393	A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( ( E. x ph /\ E. x -. ph ) -> -. E! x x = x )
1.7  Other axiomatizations related to classical predicate calculus
1.7.1  Aristotelian logic: Assertic syllogisms Model the Aristotelian assertic syllogisms using modern notation. This section shows that the Aristotelian assertic syllogisms can be proven with our axioms of logic, and also provides generally useful theorems. In antiquity Aristotelian logic and Stoic logic (see mptnan 1653) were the leading logical systems. Aristotelian logic became the leading system in medieval Europe; this section models this system (including later refinements to it). Aristotle defined syllogisms very generally ("a discourse in which certain (specific) things having been supposed, something different from the things supposed results of necessity because these things are so") Aristotle, Prior Analytics 24b18-20. However, in Prior Analytics he limits himself to categorical syllogisms that consist of three categorical propositions with specific structures. The syllogisms are the valid subset of the possible combinations of these structures. The medieval schools used vowels to identify the types of terms (a=all, e=none, i=some, and o=some are not), and named the different syllogisms with Latin words that had the vowels in the intended order. "There is a surprising amount of scholarly debate about how best to formalize Aristotle's syllogisms..." according to Aristotle's Modal Proofs: Prior Analytics A8-22 in Predicate Logic, Adriane Rini, Springer, 2011, ISBN 978-94-007-0049-9, page 28. For example, Lukasiewicz believes it is important to note that "Aristotle does not introduce singular terms or premisses into his system". Lukasiewicz also believes that Aristotelian syllogisms are predicates (having a true/false value), not inference rules: "The characteristic sign of an inference is the word 'therefore'... no syllogism is formulated by Aristotle primarily as an inference, but they are all implications." Jan Lukasiewicz, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, Second edition, Oxford, 1957, page 1-2. Lukasiewicz devised a specialized prefix notation for representing Aristotelian syllogisms instead of using standard predicate logic notation. We instead translate each Aristotelian syllogism into an inference rule, and each rule is defined using standard predicate logic notation and predicates. The predicates are represented by wff variables that may depend on the quantified variable x. Our translation is essentially identical to the one use in Rini page 18, Table 2 "Non-Modal Syllogisms in Lower Predicate Calculus (LPC)", which uses standard predicate logic with predicates. Rini states, "the crucial point is that we capture the meaning Aristotle intends, and the method by which we represent that meaning is less important." There are two differences: we make the existence criteria explicit, and we use ph, ps, and ch in the order they appear (a common Metamath convention). Patzig also uses standard predicate logic notation and predicates (though he interprets them as conditional propositions, not as inference rules); see Gunther Patzig, Aristotle's Theory of the Syllogism second edition, 1963, English translation by Jonathan Barnes, 1968, page 38. Terms such as "all" and "some" are translated into predicate logic using the approach devised by Frege and Russell. "Frege (and Russell) devised an ingenious procedure for regimenting binary quantifiers like "every" and "some" in terms of unary quantifiers like "everything" and "something": they formalized sentences of the form "Some A is B" and "Every A is B" as exists x (Ax and Bx) and all x (Ax implies Bx), respectively." "Quantifiers and Quantification", Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/quantification/. See Principia Mathematica page 22 and *10 for more information (especially *10.3 and *10.26). Expressions of the form "no ph is ps " are consistently translated as A. x ( ph -> -. ps ). These can also be expressed as -. E. x ( ph /\ ps ), per alinexa 1715. We translate "all ph is ps " to A. x ( ph -> ps ), "some ph is ps " to E. x ( ph /\ ps ), and "some ph is not ps " to E. x ( ph /\ -. ps ). It is traditional to use the singular verb "is", not the plural verb "are", in the generic expressions. By convention the major premise is listed first. In traditional Aristotelian syllogisms the predicates have a restricted form ("x is a ..."); those predicates could be modeled in modern notation by more specific constructs such as x = A, x e. A, or x C_ A. Here we use wff variables instead of specialized restricted forms. This generalization makes the syllogisms more useful in more circumstances. In addition, these expressions make it clearer that the syllogisms of Aristotelian logic are the forerunners of predicate calculus. If we used restricted forms like x e. A instead, we would not only unnecessarily limit their use, but we would also need to use set and class axioms, making their relationship to predicate calculus less clear. Using such specific constructs would also be anti-historical; Aristotle and others who directly followed his work focused on relating wholes to their parts, an approach now called part-whole theory. The work of Cantor and Peano (over 2,000 years later) led to a sharper distinction between inclusion ( C_) and membership ( e.); this distinction was not directly made in Aristotle's work. There are some widespread misconceptions about the existential assumptions made by Aristotle (aka "existential import"). Aristotle was not trying to develop something exactly corresponding to modern logic. Aristotle devised "a companion-logic for science. He relegates fictions like fairy godmothers and mermaids and unicorns to the realms of poetry and literature. In his mind, they exist outside the ambit of science. This is why he leaves no room for such non-existent entities in his logic. This is a thoughtful choice, not an inadvertent omission. Technically, Aristotelian science is a search for definitions, where a definition is "a phrase signifying a thing's essence." (Topics, I.5.102a37, Pickard-Cambridge.)... Because non-existent entities cannot be anything, they do not, in Aristotle's mind, possess an essence... This is why he leaves no place for fictional entities like goat-stags (or unicorns)." Source: Louis F. Groarke, "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy (A Peer-Reviewed Academic Resource), http://www.iep.utm.edu/aris-log/. Thus, some syllogisms have "extra" existence hypotheses that do not directly appear in Aristotle's original materials (since they were always assumed); they are added where they are needed. This affects barbari 2398, celaront 2399, cesaro 2404, camestros 2405, felapton 2410, darapti 2411, calemos 2415, fesapo 2416, and bamalip 2417. These are only the assertic syllogisms. Aristotle also defined modal syllogisms that deal with modal qualifiers such as "necessarily" and "possibly". Historically Aristotelian modal syllogisms were not as widely used. For more about modal syllogisms in a modern context, see Rini as well as Aristotle's Modal Syllogistic by Marko Malink, Harvard University Press, November 2013. We do not treat them further here. Aristotelian logic is essentially the forerunner of predicate calculus (as well as set theory since it discusses membership in groups), while Stoic logic is essentially the forerunner of propositional calculus.
Theorem	barbara 2394	"Barbara", one of the fundamental syllogisms of Aristotelian logic. All ph is ps, and all ch is ph, therefore all ch is ps. (In Aristotelian notation, AAA-1: MaP and SaM therefore SaP.) For example, given "All men are mortal" and "Socrates is a man", we can prove "Socrates is mortal". If H is the set of men, M is the set of mortal beings, and S is Socrates, these word phrases can be represented as A. x ( x e. H -> x e. M ) (all men are mortal) and A. x ( x = S -> x e. H ) (Socrates is a man) therefore A. x ( x = S -> x e. M ) (Socrates is mortal). Russell and Whitehead note that the "syllogism in Barbara is derived..." from syl 17. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1759. There are a legion of sources for Barbara, including http://www.friesian.com/aristotl.htm, http://plato.stanford.edu/entries/aristotle-logic/, and https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)
|- A. x ( ph -> ps )   &    |- A. x ( ch -> ph )   =>    |- A. x ( ch -> ps )
Theorem	celarent 2395	"Celarent", one of the syllogisms of Aristotelian logic. No ph is ps, and all ch is ph, therefore no ch is ps. (In Aristotelian notation, EAE-1: MeP and SaM therefore SeP.) For example, given the "No reptiles have fur" and "All snakes are reptiles", therefore "No snakes have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
|- A. x ( ph -> -. ps )   &    |- A. x ( ch -> ph )   =>    |- A. x ( ch -> -. ps )
Theorem	darii 2396	"Darii", one of the syllogisms of Aristotelian logic. All ph is ps, and some ch is ph, therefore some ch is ps. (In Aristotelian notation, AII-1: MaP and SiM therefore SiP.) For example, given "All rabbits have fur" and "Some pets are rabbits", therefore "Some pets have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)
|- A. x ( ph -> ps )   &    |- E. x ( ch /\ ph )   =>    |- E. x ( ch /\ ps )
Theorem	ferio 2397	"Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No ph is ps, and some ch is ph, therefore some ch is not ps. (In Aristotelian notation, EIO-1: MeP and SiM therefore SoP.) For example, given "No homework is fun" and "Some reading is homework", therefore "Some reading is not fun". This is essentially a logical axiom in Aristotelian logic. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
|- A. x ( ph -> -. ps )   &    |- E. x ( ch /\ ph )   =>    |- E. x ( ch /\ -. ps )
Theorem	barbari 2398	"Barbari", one of the syllogisms of Aristotelian logic. All ph is ps, all ch is ph, and some ch exist, therefore some ch is ps. (In Aristotelian notation, AAI-1: MaP and SaM therefore SiP.) For example, given "All men are mortal", "All Greeks are men", and "Greeks exist", therefore "Some Greeks are mortal". Note the existence hypothesis (to prove the "some" in the conclusion). Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 30-Aug-2016.)
|- A. x ( ph -> ps )   &    |- A. x ( ch -> ph )   &    |- E. x ch   =>    |- E. x ( ch /\ ps )
Theorem	celaront 2399	"Celaront", one of the syllogisms of Aristotelian logic. No ph is ps, all ch is ph, and some ch exist, therefore some ch is not ps. (In Aristotelian notation, EAO-1: MeP and SaM therefore SoP.) For example, given "No reptiles have fur", "All snakes are reptiles.", and "Snakes exist.", prove "Some snakes have no fur". Note the existence hypothesis. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
|- A. x ( ph -> -. ps )   &    |- A. x ( ch -> ph )   &    |- E. x ch   =>    |- E. x ( ch /\ -. ps )
Theorem	cesare 2400	"Cesare", one of the syllogisms of Aristotelian logic. No ph is ps, and all ch is ps, therefore no ch is ph. (In Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to celarent 2395. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.)
|- A. x ( ph -> -. ps )   &    |- A. x ( ch -> ps )   =>    |- A. x ( ch -> -. ph )