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
Theorem	nfeu 2301	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 2302	Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)
|- F/ x ph   =>    |- F/ x E* y ph
Theorem	eubid 2303	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 2304	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 2305*	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 2306*	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 2307	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 2308	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 2309	Existential uniqueness implies existence. For a shorter proof using more axioms, see euexALT 2329. (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 2310	Something exists or at most one exists. (Contributed by NM, 8-Mar-1995.)
|- ( E. x ph \/ E* x ph )
Theorem	eu5 2311	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 2312	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 2313*	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 2314	Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.)
|- ( E! x ph -> E* x ph )
Theorem	eumoi 2315	"At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
|- E! x ph   =>    |- E* x ph
Theorem	moabs 2316	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 2317	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	eumo0OLD 2318*	Obsolete as of 6-Jun-2018. Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.) (Proof modification is discouraged.)
|- F/ y ph   =>    |- ( E! x ph -> E. y A. x ( ph -> x = y ) )
Theorem	sb8eu 2319	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	sb8euOLD 2320	Obsolete proof of sb8eu 2319 as of 24-Aug-2019. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 8-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
|- F/ y ph   =>    |- ( E! x ph <-> E! y [ y / x ] ph )
Theorem	sb8mo 2321	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 2322	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 2323	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 2324*	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	mo3OLD 2325*	Obsolete proof of mo3 2324 as of 20-Jul-2019. (Contributed by NM, 8-Mar-1995.) (Revised by Wolf Lammen, 3-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
|- F/ y ph   =>    |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
Theorem	mo 2326*	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 2327*	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 2328*	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 2329	Alternate proof of euex 2309. 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	eu3OLD 2330*	Obsolete theorem as of 29-May-2018. Superseded by eu3v 2313 that better fits common usage pattern. (Contributed by NM, 8-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
|- F/ y ph   =>    |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
Theorem	eu5OLD 2331	Obsolete proof of eu5 2311 as of 25-May-2019 (Contributed by NM, 23-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
Theorem	euor 2332	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 2333*	Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
|- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) )
Theorem	euor2 2334	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	mo2OLD 2335*	Obsolete proof of mo2 2294 as of 28-May-2019. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
|- F/ y ph   =>    |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
Theorem	sbmo 2336*	Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( [ y / x ] E* z ph <-> E* z [ y / x ] ph )
Theorem	mo4f 2337*	"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 2338*	"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 2339*	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 2340	"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 2341	"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 2342	TODO-NM: AV and BJ propose to keep this theorem as morim . Obsolete theorem as of 22-Dec-2018. Use moimi 2341 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 2343	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 2344	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 2345	"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 2346	"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 2347	"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 2348	"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 2349	"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 2350	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 2351	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 2352*	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 2353	Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.)
|- E* x ( ph /\ E* x ph )
Theorem	moaneu 2354	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 2355*	Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
|- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) )
Theorem	mopick 2356	"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	mopickOLD 2357	Obsolete proof of mopick 2356 as of 15-Sep-2019. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 29-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
Theorem	eupick 2358	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 2359	Version of eupick 2358 with closed formulas. (Contributed by NM, 6-Sep-2008.)
|- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> A. x ( ph -> ps ) )
Theorem	eupickb 2360	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 2361	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 2362	"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 1680. (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 2363	"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 2364*	"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 2365	Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)
|- ( E* x E. y ph -> A. y E* x ph )
Theorem	2euex 2366	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 2367	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 2368	Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
|- ( E! x E! y ph -> E. x E. y ph )
Theorem	2moswap 2369	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 2370	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 2371	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 2372*	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 2380 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 2373*	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	2moOLD 2374*	Obsolete proof of 2mo 2373 as of 2-Nov-2019. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 25-Oct-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( 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 2375*	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 2376	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	2eu1OLD 2377	Obsolete proof of 2eu1 2376 as of 11-Nov-2019. (Contributed by NM, 3-Dec-2001.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( A. x E* y ph -> ( E! x E! y ph <-> ( E! x E. y ph /\ E! y E. x ph ) ) )
Theorem	2eu2 2378	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 2379	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 2380*	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 2376 for a condition under which the naive definition holds and 2exeu 2371 for a one-way implication. See 2eu5 2382 and 2eu8 2386 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	2eu4OLD 2381*	Obsolete proof of 2eu4 2380 as of 14-Sep-2019. (Contributed by NM, 3-Dec-2001.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( 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 2382*	An alternate definition of double existential uniqueness (see 2eu4 2380). 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 2383*	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	2eu6OLD 2384*	Obsolete proof of 2eu6 2383 as of 21-Sep-2019. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( 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 2385	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 2386	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 2385. (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	euequ1OLD 2387*	Obsolete proof of euequ1 2289 as of 8-Sep-2019. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
|- E! x x = y
Theorem	exists1 2388*	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 4647. (Contributed by NM, 5-Apr-2004.)
|- ( E! x x = x <-> A. x x = y )
Theorem	exists2 2389	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 )
1.8  Other axiomatizations related to classical predicate calculus
1.8.1  Predicate calculus with all distinct variables
Axiom	ax-7d 2390*	Distinct variable version of ax-11 1843. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( A. x A. y ph -> A. y A. x ph )
Axiom	ax-8d 2391*	Distinct variable version of ax-7 1791. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( x = y -> ( x = z -> y = z ) )
Axiom	ax-9d1 2392	Distinct variable version of ax-6 1748, equal variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- -. A. x -. x = x
Axiom	ax-9d2 2393*	Distinct variable version of ax-6 1748, distinct variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- -. A. x -. x = y
Axiom	ax-10d 2394*	Distinct variable version of axc11n 2050. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( A. x x = y -> A. y y = x )
Axiom	ax-11d 2395*	Distinct variable version of ax-12 1855. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
1.8.2  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 1601) 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 1664. 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 2400, celaront 2401, cesaro 2406, camestros 2407, felapton 2412, darapti 2413, calemos 2417, fesapo 2418, and bamalip 2419. 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 2396	"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 16. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1704. 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 2397	"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 2398	"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 2399	"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 2400	"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 )