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	sb8mo 2301	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 2302	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 2303	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 2304*	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 2305*	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 2306*	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 2307*	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 2308	Alternate proof of euex 2291. 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 2309	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 2310*	Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
|- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) )
Theorem	euor2 2311	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 2312*	Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( [ y / x ] E* z ph <-> E* z [ y / x ] ph )
Theorem	mo4f 2313*	"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 2314*	"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 2315*	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 2316	"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 2317	"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 2318	TODO-NM: AV and BJ propose to keep this theorem as morim . Obsolete theorem as of 22-Dec-2018. Use moimi 2317 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 2319	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 2320	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 2321	"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 2322	"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 2323	"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 2324	"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 2325	"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 2326	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 2327	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 2328*	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 2329	Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.)
|- E* x ( ph /\ E* x ph )
Theorem	moaneu 2330	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 2331*	Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
|- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) )
Theorem	mopick 2332	"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 2333	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 2334	Version of eupick 2333 with closed formulas. (Contributed by NM, 6-Sep-2008.)
|- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> A. x ( ph -> ps ) )
Theorem	eupickb 2335	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 2336	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 2337	"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 1725. (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 2338	"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 2339*	"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 2340	Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)
|- ( E* x E. y ph -> A. y E* x ph )
Theorem	2euex 2341	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 2342	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 2343	Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
|- ( E! x E! y ph -> E. x E. y ph )
Theorem	2moswap 2344	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 2345	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 2346	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 2347*	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 2355 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 2348*	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 2349*	Obsolete proof of 2mo 2348 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 2350*	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 2351	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 2352	Obsolete proof of 2eu1 2351 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 2353	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 2354	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 2355*	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 2351 for a condition under which the naive definition holds and 2exeu 2346 for a one-way implication. See 2eu5 2356 and 2eu8 2359 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 2356*	An alternate definition of double existential uniqueness (see 2eu4 2355). 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 2357*	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 2358	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 2359	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 2358. (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 2360*	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 4613. (Contributed by NM, 5-Apr-2004.)
|- ( E! x x = x <-> A. x x = y )
Theorem	exists2 2361	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 1648) 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 1709. 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 2366, celaront 2367, cesaro 2372, camestros 2373, felapton 2378, darapti 2379, calemos 2383, fesapo 2384, and bamalip 2385. 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 2362	"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 1748. 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 2363	"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 2364	"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 2365	"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 2366	"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 2367	"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 2368	"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 2363. (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 )
Theorem	camestres 2369	"Camestres", one of the syllogisms of Aristotelian logic. All ph is ps, and no ch is ps, therefore no ch is ph. (In Aristotelian notation, AEE-2: PaM and SeM therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
|- A. x ( ph -> ps )   &    |- A. x ( ch -> -. ps )   =>    |- A. x ( ch -> -. ph )
Theorem	festino 2370	"Festino", one of the syllogisms of Aristotelian logic. No ph is ps, and some ch is ps, therefore some ch is not ph. (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.)
|- A. x ( ph -> -. ps )   &    |- E. x ( ch /\ ps )   =>    |- E. x ( ch /\ -. ph )
Theorem	baroco 2371	"Baroco", one of the syllogisms of Aristotelian logic. All ph is ps, and some ch is not ps, therefore some ch is not ph. (In Aristotelian notation, AOO-2: PaM and SoM therefore SoP.) For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative." (Contributed by David A. Wheeler, 28-Aug-2016.)
|- A. x ( ph -> ps )   &    |- E. x ( ch /\ -. ps )   =>    |- E. x ( ch /\ -. ph )
Theorem	cesaro 2372	"Cesaro", one of the syllogisms of Aristotelian logic. No ph is ps, all ch is ps, and ch exist, therefore some ch is not ph. (In Aristotelian notation, EAO-2: PeM and SaM therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
|- A. x ( ph -> -. ps )   &    |- A. x ( ch -> ps )   &    |- E. x ch   =>    |- E. x ( ch /\ -. ph )
Theorem	camestros 2373	"Camestros", one of the syllogisms of Aristotelian logic. All ph is ps, no ch is ps, and ch exist, therefore some ch is not ph. (In Aristotelian notation, AEO-2: PaM and SeM therefore SoP.) For example, "All horses have hooves", "No humans have hooves", and humans exist, therefore "Some humans are not horses". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
|- A. x ( ph -> ps )   &    |- A. x ( ch -> -. ps )   &    |- E. x ch   =>    |- E. x ( ch /\ -. ph )
Theorem	datisi 2374	"Datisi", one of the syllogisms of Aristotelian logic. All ph is ps, and some ph is ch, therefore some ch is ps. (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
|- A. x ( ph -> ps )   &    |- E. x ( ph /\ ch )   =>    |- E. x ( ch /\ ps )
Theorem	disamis 2375	"Disamis", one of the syllogisms of Aristotelian logic. Some ph is ps, and all ph is ch, therefore some ch is ps. (In Aristotelian notation, IAI-3: MiP and MaS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
|- E. x ( ph /\ ps )   &    |- A. x ( ph -> ch )   =>    |- E. x ( ch /\ ps )
Theorem	ferison 2376	"Ferison", one of the syllogisms of Aristotelian logic. No ph is ps, and some ph is ch, therefore some ch is not ps. (In Aristotelian notation, EIO-3: MeP and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
|- A. x ( ph -> -. ps )   &    |- E. x ( ph /\ ch )   =>    |- E. x ( ch /\ -. ps )
Theorem	bocardo 2377	"Bocardo", one of the syllogisms of Aristotelian logic. Some ph is not ps, and all ph is ch, therefore some ch is not ps. (In Aristotelian notation, OAO-3: MoP and MaS therefore SoP.) For example, "Some cats have no tails", "All cats are mammals", therefore "Some mammals have no tails". A reorder of disamis 2375; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.)
|- E. x ( ph /\ -. ps )   &    |- A. x ( ph -> ch )   =>    |- E. x ( ch /\ -. ps )
Theorem	felapton 2378	"Felapton", one of the syllogisms of Aristotelian logic. No ph is ps, all ph is ch, and some ph exist, therefore some ch is not ps. (In Aristotelian notation, EAO-3: MeP and MaS therefore SoP.) For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
|- A. x ( ph -> -. ps )   &    |- A. x ( ph -> ch )   &    |- E. x ph   =>    |- E. x ( ch /\ -. ps )
Theorem	darapti 2379	"Darapti", one of the syllogisms of Aristotelian logic. All ph is ps, all ph is ch, and some ph exist, therefore some ch is ps. (In Aristotelian notation, AAI-3: MaP and MaS therefore SiP.) For example, "All squares are rectangles" and "All squares are rhombuses", therefore "Some rhombuses are rectangles". (Contributed by David A. Wheeler, 28-Aug-2016.)
|- A. x ( ph -> ps )   &    |- A. x ( ph -> ch )   &    |- E. x ph   =>    |- E. x ( ch /\ ps )
Theorem	calemes 2380	"Calemes", one of the syllogisms of Aristotelian logic. All ph is ps, and no ps is ch, therefore no ch is ph. (In Aristotelian notation, AEE-4: PaM and MeS therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
|- A. x ( ph -> ps )   &    |- A. x ( ps -> -. ch )   =>    |- A. x ( ch -> -. ph )
Theorem	dimatis 2381	"Dimatis", one of the syllogisms of Aristotelian logic. Some ph is ps, and all ps is ch, therefore some ch is ph. (In Aristotelian notation, IAI-4: PiM and MaS therefore SiP.) For example, "Some pets are rabbits.", "All rabbits have fur", therefore "Some fur bearing animals are pets". Like darii 2364 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)
|- E. x ( ph /\ ps )   &    |- A. x ( ps -> ch )   =>    |- E. x ( ch /\ ph )
Theorem	fresison 2382	"Fresison", one of the syllogisms of Aristotelian logic. No ph is ps (PeM), and some ps is ch (MiS), therefore some ch is not ph (SoP). (In Aristotelian notation, EIO-4: PeM and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
|- A. x ( ph -> -. ps )   &    |- E. x ( ps /\ ch )   =>    |- E. x ( ch /\ -. ph )
Theorem	calemos 2383	"Calemos", one of the syllogisms of Aristotelian logic. All ph is ps (PaM), no ps is ch (MeS), and ch exist, therefore some ch is not ph (SoP). (In Aristotelian notation, AEO-4: PaM and MeS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
|- A. x ( ph -> ps )   &    |- A. x ( ps -> -. ch )   &    |- E. x ch   =>    |- E. x ( ch /\ -. ph )
Theorem	fesapo 2384	"Fesapo", one of the syllogisms of Aristotelian logic. No ph is ps, all ps is ch, and ps exist, therefore some ch is not ph. (In Aristotelian notation, EAO-4: PeM and MaS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
|- A. x ( ph -> -. ps )   &    |- A. x ( ps -> ch )   &    |- E. x ps   =>    |- E. x ( ch /\ -. ph )
Theorem	bamalip 2385	"Bamalip", one of the syllogisms of Aristotelian logic. All ph is ps, all ps is ch, and ph exist, therefore some ch is ph. (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 2366. (Contributed by David A. Wheeler, 28-Aug-2016.)
|- A. x ( ph -> ps )   &    |- A. x ( ps -> ch )   &    |- E. x ph   =>    |- E. x ( ch /\ ph )
1.7.2  Intuitionistic logic Intuitionistic (constructive) logic is similar to classical logic with the notable omission of ax-3 8 and theorems such as exmid 417 or peirce 185. We mostly treat intuitionistic logic in a separate file, iset.mm, which is known as the Intuitionistic Logic Explorer on the web site. However, iset.mm has a number of additional axioms (mainly to replace definitions like df-or 372 and df-ex 1661 which are not valid in intuitionistic logic) and we want to prove those axioms here to demonstrate that adding those axioms in iset.mm does not make iset.mm any less consistent than set.mm. The following axioms are unchanged between set.mm and iset.mm: ax-1 6, ax-2 7, ax-mp 5, ax-4 1679, ax-11 1893, ax-gen 1666, ax-7 1840, ax-12 1906, ax-8 1871, ax-9 1873, and ax-5 1749. In this list of axioms, the ones that repeat earlier theorems are marked "(New usage is discouraged.)" so that the earlier theorems will be used consistently in other proofs.
Theorem	axia1 2386	Left 'and' elimination (intuitionistic logic axiom ax-ia1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
|- ( ( ph /\ ps ) -> ph )
Theorem	axia2 2387	Right 'and' elimination (intuitionistic logic axiom ax-ia2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
|- ( ( ph /\ ps ) -> ps )
Theorem	axia3 2388	'And' introduction (intuitionistic logic axiom ax-ia3). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
|- ( ph -> ( ps -> ( ph /\ ps ) ) )
Theorem	axin1 2389	'Not' introduction (intuitionistic logic axiom ax-in1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
|- ( ( ph -> -. ph ) -> -. ph )
Theorem	axin2 2390	'Not' elimination (intuitionistic logic axiom ax-in2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
|- ( -. ph -> ( ph -> ps ) )
Theorem	axio 2391	Definition of 'or' (intuitionistic logic axiom ax-io). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
|- ( ( ( ph \/ ch ) -> ps ) <-> ( ( ph -> ps ) /\ ( ch -> ps ) ) )
Theorem	axi4 2392	Specialization (intuitionistic logic axiom ax-4). This is just sp 1911 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
|- ( A. x ph -> ph )
Theorem	axi5r 2393	Converse of ax-c4 (intuitionistic logic axiom ax-i5r). (Contributed by Jim Kingdon, 31-Dec-2017.)
|- ( ( A. x ph -> A. x ps ) -> A. x ( A. x ph -> ps ) )
Theorem	axial 2394	x is not free in A. x ph (intuitionistic logic axiom ax-ial). (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
|- ( A. x ph -> A. x A. x ph )
Theorem	axie1 2395	x is not free in E. x ph (intuitionistic logic axiom ax-ie1). (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
|- ( E. x ph -> A. x E. x ph )
Theorem	axie2 2396	A key property of existential quantification (intuitionistic logic axiom ax-ie2). (Contributed by Jim Kingdon, 31-Dec-2017.)
|- ( A. x ( ps -> A. x ps ) -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )
Theorem	axi9 2397	Axiom of existence (intuitionistic logic axiom ax-i9). In classical logic, this is equivalent to ax-6 1795 but in intuitionistic logic it needs to be stated using the existential quantifier. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
|- E. x x = y
Theorem	axi10 2398	Axiom of Quantifier Substitution (intuitionistic logic axiom ax-10). This is just axc11n 2105 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
|- ( A. x x = y -> A. y y = x )
Theorem	axi12 2399	Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12). In classical logic, this is mostly a restatement of axc9 2102 (with one additional quantifier). But in intuitionistic logic, changing the negations and implications to disjunctions makes it stronger. (Contributed by Jim Kingdon, 31-Dec-2017.)
|- ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )
Theorem	axbnd 2400	Axiom of Bundling (intuitionistic logic axiom ax-bnd). In classical logic, this and axi12 2399 are fairly straightforward consequences of axc9 2102. But in intuitionistic logic, it is not easy to add the extra A. x to axi12 2399 and so we treat the two as separate axioms. (Contributed by Jim Kingdon, 22-Mar-2018.)
|- ( A. z z = x \/ ( A. z z = y \/ A. x A. z ( x = y -> A. z x = y ) ) )