HomeHome Metamath Proof Explorer
Theorem List (p. 23 of 313)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21423)
  Hilbert Space Explorer  Hilbert Space Explorer
(21424-22946)
  Users' Mathboxes  Users' Mathboxes
(22947-31284)
 

Theorem List for Metamath Proof Explorer - 2201-2300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2eu6 2201* Two equivalent expressions for double existential uniqueness. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  E. z E. w A. x A. y ( ph  <->  ( x  =  z  /\  y  =  w )
 ) )
 
Theorem2eu7 2202 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 ) )
 
Theorem2eu8 2203 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 2202. (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 ) )
 
Theoremeuequ1 2204* Equality has existential uniqueness. Special case of eueq1 2889 proved using only predicate calculus. (Contributed by Stefan Allan, 4-Dec-2008.)
 |- 
 E! x  x  =  y
 
Theoremexists1 2205* 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 4139. (Contributed by NM, 5-Apr-2004.)
 |-  ( E! x  x  =  x  <->  A. x  x  =  y )
 
Theoremexists2 2206 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  Predicate calculus with all distinct variables
 
Axiomax-7d 2207* Distinct varable version of ax-7 1535. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Axiomax-8d 2208* Distinct varable version of ax-8 1623. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( x  =  y 
 ->  ( x  =  z 
 ->  y  =  z
 ) )
 
Axiomax-9d1 2209 Distinct varable version of ax-9 1684, variables equal case. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 -.  A. x  -.  x  =  x
 
Axiomax-9d2 2210* Distinct varable version of ax-9 1684, variables distinct case. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 -.  A. x  -.  x  =  y
 
Axiomax-10d 2211* Distinct varable version of ax-10 1678. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Axiomax-11d 2212* Distinct varable version of ax-11 1624. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( x  =  y 
 ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
 ) )
 
1.7.2  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 mpto1 1528) 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 aproach 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 1576. 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 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 Aristolean 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.

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 2217, celaront 2218, cesaro 2223, camestros 2224, felapton 2229, darapti 2230, calemos 2234, fesapo 2235, and bamalip 2236.

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.

Aristotelean 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.

 
Theorembarbara 2213 "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 1617. 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 )
 
Theoremcelarent 2214 "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 )
 
Theoremdarii 2215 "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 )
 
Theoremferio 2216 "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 )
 
Theorembarbari 2217 "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 )
 
Theoremcelaront 2218 "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 )
 
Theoremcesare 2219 "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 2214. (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 )
 
Theoremcamestres 2220 "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 )
 
Theoremfestino 2221 "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 )
 
Theorembaroco 2222 "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 )
 
Theoremcesaro 2223 "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 )
 
Theoremcamestros 2224 "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 )
 
Theoremdatisi 2225 "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 )
 
Theoremdisamis 2226 "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 )
 
Theoremferison 2227 "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 )
 
Theorembocardo 2228 "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 2226; 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 )
 
Theoremfelapton 2229 "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 )
 
Theoremdarapti 2230 "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 )
 
Theoremcalemes 2231 "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 )
 
Theoremdimatis 2232 "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 2215 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)
 |- 
 E. x ( ph  /\ 
 ps )   &    |-  A. x ( ps  ->  ch )   =>    |-  E. x ( ch  /\  ph )
 
Theoremfresison 2233 "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 )
 
Theoremcalemos 2234 "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 )
 
Theoremfesapo 2235 "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 )
 
Theorembamalip 2236 "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 2217. (Contributed by David A. Wheeler, 28-Aug-2016.)
 |- 
 A. x ( ph  ->  ps )   &    |-  A. x ( ps  ->  ch )   &    |-  E. x ph   =>    |-  E. x ( ch  /\  ph )
 
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY

Set theory uses the formalism of propositional and predicate calculus to assert properties of arbitrary mathematical objects called "sets." A set can be contained in another set, and this relationship is indicated by the  e. symbol. Starting with the simplest mathematical object, called the empty set, set theory builds up more and more complex structures whose existence follows from the axioms, eventually resulting in extremely complicated sets that we identify with the real numbers and other familiar mathematical objects.

A simplistic concept of sets, sometimes called "naive set theory", is vulnerable to a paradox called "Russell's paradox" (ru 2934), a discovery that revolutionized the foundations of mathematics and logic. Russell's Paradox spawned the development of set theories that countered the paradox, including the ZF set theory that is most widely used and is defined here.

Except for Extensionality, the axioms basically say, "given an arbitrary set x (and, in the cases of Replacement and Regularity, provided that an antecedent is satisfied), there exists another set y based on or constructed from it, with the stated properties." (The axiom of Extensionality can also be restated this way as shown by axext2 2238.) The individual axiom links provide more detailed descriptions. We derive the redundant ZF axioms of Separation, Null Set, and Pairing from the others as theorems.

 
2.1  ZF Set Theory - start with the Axiom of Extensionality
 
2.1.1  Introduce the Axiom of Extensionality
 
Axiomax-ext 2237* Axiom of Extensionality. An axiom of Zermelo-Fraenkel set theory. It states that two sets are identical if they contain the same elements. Axiom Ext of [BellMachover] p. 461.

Set theory can also be formulated with a single primitive predicate  e. on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes  ( A. w
( w  e.  x  <->  w  e.  y )  -> 
( x  e.  z  ->  y  e.  z ) ), and equality  x  =  y is defined as  A. w ( w  e.  x  <->  w  e.  y
). All of the usual axioms of equality then become theorems of set theory. See, for example, Axiom 1 of [TakeutiZaring] p. 8.

To use the above "equality-free" version of Extensionality with Metamath's logical axioms, we would rewrite ax-8 1623 through ax-16 1927 with equality expanded according to the above definition. Some of those axioms could be proved from set theory and would be redundant. Not all of them are redundant, since our axioms of predicate calculus make essential use of equality for the proper substitution that is a primitive notion in traditional predicate calculus. A study of such an axiomatization would be an interesting project for someone exploring the foundations of logic.

General remarks: Our set theory axioms are presented using defined connectives (
<->, 
E., etc.) for convenience. However, it is implicitly understood that the actual axioms use only the primitive connectives  ->,  -.,  A.,  =, and  e.. It is straightforward to establish the equivalence between the actual axioms and the ones we display, and we will not do so.

It is important to understand that strictly speaking, all of our set theory axioms are really schemes that represent an infinite number of actual axioms. This is inherent in the design of Metamath ("metavariable math"), which manipulates only metavariables. For example, the metavariable  x in ax-ext 2237 can represent any actual variable v1, v2, v3,... . Distinct variable restrictions ($d) prevent us from substituting say v1 for both  x and  z. This is in contrast to typical textbook presentations that present actual axioms (except for Replacement ax-rep 4071, which involves a wff metavariable). In practice, though, the theorems and proofs are essentially the same. The $d restrictions make each of the infinite axioms generated by the ax-ext 2237 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 5-Aug-1993.)

 |-  ( A. z ( z  e.  x  <->  z  e.  y
 )  ->  x  =  y )
 
Theoremaxext2 2238* The Axiom of Extensionality (ax-ext 2237) restated so that it postulates the existence of a set  z given two arbitrary sets 
x and  y. This way to express it follows the general idea of the other ZFC axioms, which is to postulate the existence of sets given other sets. (Contributed by NM, 28-Sep-2003.)
 |- 
 E. z ( ( z  e.  x  <->  z  e.  y
 )  ->  x  =  y )
 
Theoremaxext3 2239* A generalization of the Axiom of Extensionality in which  x and  y need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |-  ( A. z ( z  e.  x  <->  z  e.  y
 )  ->  x  =  y )
 
Theoremaxext4 2240* A bidirectional version of Extensionality. Although this theorem "looks" like it is just a definition of equality, it requires the Axiom of Extensionality for its proof under our axiomatization. See the comments for ax-ext 2237 and df-cleq 2249. (Contributed by NM, 14-Nov-2008.)
 |-  ( x  =  y  <->  A. z ( z  e.  x  <->  z  e.  y
 ) )
 
Theorembm1.1 2241* Any set defined by a property is the only set defined by that property. Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM, 30-Jun-1994.)
 |- 
 F/ x ph   =>    |-  ( E. x A. y ( y  e.  x  <->  ph )  ->  E! x A. y ( y  e.  x  <->  ph ) )
 
2.1.2  Class abstractions (a.k.a. class builders)
 
Syntaxcab 2242 Introduce the class builder or class abstraction notation ("the class of sets  x such that  ph is true"). Our class variables  A,  B, etc. range over class builders (implicitly in the case of defined class terms such as df-nul 3398). Note that a set variable can be expressed as a class builder per theorem cvjust 2251, justifying the assignment of set variables to class variables via the use of cv 1618.
 class  { x  |  ph }
 
Definitiondf-clab 2243 Define class abstraction notation (so-called by Quine), also called a "class builder" in the literature.  x and  y need not be distinct. Definition 2.1 of [Quine] p. 16. Typically,  ph will have  y as a free variable, and " { y  |  ph } " is read "the class of all sets  y such that  ph ( y ) is true." We do not define  { y  |  ph } in isolation but only as part of an expression that extends or "overloads" the  e. relationship.

This is our first use of the 
e. symbol to connect classes instead of sets. The syntax definition wcel 1621, which extends or "overloads" the wel 1622 definition connecting set variables, requires that both sides of  e. be a class. In df-cleq 2249 and df-clel 2252, we introduce a new kind of variable (class variable) that can substituted with expressions such as  { y  | 
ph }. In the present definition, the  x on the left-hand side is a set variable. Syntax definition cv 1618 allows us to substitute a set variable  x for a class variable: all sets are classes by cvjust 2251 (but not necessarily vice-versa). For a full description of how classes are introduced and how to recover the primitive language, see the discussion in Quine (and under abeq2 2361 for a quick overview).

Because class variables can be substituted with compound expressions and set variables cannot, it is often useful to convert a theorem containing a free set variable to a more general version with a class variable. This is done with theorems such as vtoclg 2794 which is used, for example, to convert elirrv 7244 to elirr 7245.

This is called the "axiom of class comprehension" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. He calls the construction  {
y  |  ph } a "class term".

For a general discussion of the theory of classes, see http://us.metamath.org/mpegif/mmset.html#class. (Contributed by NM, 5-Aug-1993.)

 |-  ( x  e.  {
 y  |  ph }  <->  [ x  /  y ] ph )
 
Theoremabid 2244 Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  e.  { x  |  ph }  <->  ph )
 
Theoremhbab1 2245* Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.)
 |-  ( y  e.  { x  |  ph }  ->  A. x  y  e.  { x  |  ph } )
 
Theoremnfsab1 2246* Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x  y  e. 
 { x  |  ph }
 
Theoremhbab 2247* Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.)
 |-  ( ph  ->  A. x ph )   =>    |-  ( z  e.  {
 y  |  ph }  ->  A. x  z  e.  {
 y  |  ph } )
 
Theoremnfsab 2248* Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x ph   =>    |- 
 F/ x  z  e. 
 { y  |  ph }
 
Definitiondf-cleq 2249* Define the equality connective between classes. Definition 2.7 of [Quine] p. 18. Also Definition 4.5 of [TakeutiZaring] p. 13; Chapter 4 provides its justification and methods for eliminating it. Note that its elimination will not necessarily result in a single wff in the original language but possibly a "scheme" of wffs.

This is an example of a somewhat "risky" definition, meaning that it has a more complex than usual soundness justification (outside of Metamath), because it "overloads" or reuses the existing equality symbol rather than introducing a new symbol. This allows us to make statements that may not hold for the original symbol. For example, it permits us to deduce  y  =  z  <->  A. x ( x  e.  y  <->  x  e.  z
), which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see theorem axext4 2240). We therefore include this axiom as a hypothesis, so that the use of Extensionality is properly indicated.

We could avoid this complication by introducing a new symbol, say =2, in place of  =. This would also have the advantage of making elimination of the definition straightforward, so that we could eliminate Extensionality as a hypothesis. We would then also have the advantage of being able to identify in various proofs exactly where Extensionality truly comes into play rather than just being an artifact of a definition.. One of our theorems would then be  x =2  y  <->  x  =  y by invoking Extensionality.

However, to conform to literature usage, we retain this overloaded definition. This also makes some proofs shorter and probably easier to read, without the constant switching between two kinds of equality.

See also comments under df-clab 2243, df-clel 2252, and abeq2 2361.

In the form of dfcleq 2250, this is called the "axiom of extensionality" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms.

For a general discussion of the theory of classes, see http://us.metamath.org/mpegif/mmset.html#class. (Contributed by NM, 15-Sep-1993.)

 |-  ( A. x ( x  e.  y  <->  x  e.  z
 )  ->  y  =  z )   =>    |-  ( A  =  B  <->  A. x ( x  e.  A  <->  x  e.  B ) )
 
Theoremdfcleq 2250* The same as df-cleq 2249 with the hypothesis removed using the Axiom of Extensionality ax-ext 2237. (Contributed by NM, 15-Sep-1993.)
 |-  ( A  =  B  <->  A. x ( x  e.  A  <->  x  e.  B ) )
 
Theoremcvjust 2251* Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a set variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv 1618, which allows us to substitute a set variable for a class variable. See also cab 2242 and df-clab 2243. Note that this is not a rigorous justification, because cv 1618 is used as part of the proof of this theorem, but a careful argument can be made outside of the formalism of Metamath, for example as is done in Chapter 4 of Takeuti and Zaring. See also the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." (Contributed by NM, 7-Nov-2006.)
 |-  x  =  { y  |  y  e.  x }
 
Definitiondf-clel 2252* Define the membership connective between classes. Theorem 6.3 of [Quine] p. 41, or Proposition 4.6 of [TakeutiZaring] p. 13, which we adopt as a definition. See these references for its metalogical justification. Note that like df-cleq 2249 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2249 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 2063), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2243. Alternate definitions of  A  e.  B (but that require either  A or  B to be a set) are shown by clel2 2855, clel3 2857, and clel4 2858.

This is called the "axiom of membership" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms.

For a general discussion of the theory of classes, see http://us.metamath.org/mpegif/mmset.html#class. (Contributed by NM, 5-Aug-1993.)

 |-  ( A  e.  B  <->  E. x ( x  =  A  /\  x  e.  B ) )
 
Theoremeqriv 2253* Infer equality of classes from equivalence of membership. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  e.  A  <->  x  e.  B )   =>    |-  A  =  B
 
Theoremeqrdv 2254* Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.)
 |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremeqrdav 2255* Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.)
 |-  ( ( ph  /\  x  e.  A )  ->  x  e.  C )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  x  e.  C )   &    |-  ( ( ph  /\  x  e.  C )  ->  ( x  e.  A  <->  x  e.  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremeqid 2256 Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41.

This law is thought to have originated with Aristotle (Metaphysics, Book VII, Part 17). (Thanks to Stefan Allan for this information.) (Contributed by NM, 5-Aug-1993.)

 |-  A  =  A
 
Theoremeqidd 2257 Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.)
 |-  ( ph  ->  A  =  A )
 
Theoremeqcom 2258 Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  <->  B  =  A )
 
Theoremeqcoms 2259 Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  -> 
 ph )   =>    |-  ( B  =  A  -> 
 ph )
 
Theoremeqcomi 2260 Inference from commutative law for class equality. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   =>    |-  B  =  A
 
Theoremeqcomd 2261 Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  B  =  A )
 
Theoremeqeq1 2262 Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( A  =  C  <->  B  =  C ) )
 
Theoremeqeq1i 2263 Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   =>    |-  ( A  =  C 
 <->  B  =  C )
 
Theoremeqeq1d 2264 Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  =  C  <->  B  =  C ) )
 
Theoremeqeq2 2265 Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( C  =  A  <->  C  =  B ) )
 
Theoremeqeq2i 2266 Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   =>    |-  ( C  =  A 
 <->  C  =  B )
 
Theoremeqeq2d 2267 Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  =  A  <->  C  =  B ) )
 
Theoremeqeq12 2268 Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  =  C 
 <->  B  =  D ) )
 
Theoremeqeq12i 2269 A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  =  C  <->  B  =  D )
 
Theoremeqeq12d 2270 A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  =  C  <->  B  =  D ) )
 
Theoremeqeqan12d 2271 A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  C  =  D )   =>    |-  ( ( ph  /\ 
 ps )  ->  ( A  =  C  <->  B  =  D ) )
 
Theoremeqeqan12rd 2272 A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  C  =  D )   =>    |-  ( ( ps 
 /\  ph )  ->  ( A  =  C  <->  B  =  D ) )
 
Theoremeqtr 2273 Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.)
 |-  ( ( A  =  B  /\  B  =  C )  ->  A  =  C )
 
Theoremeqtr2 2274 A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ( A  =  B  /\  A  =  C )  ->  B  =  C )
 
Theoremeqtr3 2275 A transitive law for class equality. (Contributed by NM, 20-May-2005.)
 |-  ( ( A  =  C  /\  B  =  C )  ->  A  =  B )
 
Theoremeqtri 2276 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  B  =  C   =>    |-  A  =  C
 
Theoremeqtr2i 2277 An equality transitivity inference. (Contributed by NM, 21-Feb-1995.)
 |-  A  =  B   &    |-  B  =  C   =>    |-  C  =  A
 
Theoremeqtr3i 2278 An equality transitivity inference. (Contributed by NM, 6-May-1994.)
 |-  A  =  B   &    |-  A  =  C   =>    |-  B  =  C
 
Theoremeqtr4i 2279 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  C  =  B   =>    |-  A  =  C
 
Theorem3eqtri 2280 An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.)
 |-  A  =  B   &    |-  B  =  C   &    |-  C  =  D   =>    |-  A  =  D
 
Theorem3eqtrri 2281 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  A  =  B   &    |-  B  =  C   &    |-  C  =  D   =>    |-  D  =  A
 
Theorem3eqtr2i 2282 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.)
 |-  A  =  B   &    |-  C  =  B   &    |-  C  =  D   =>    |-  A  =  D
 
Theorem3eqtr2ri 2283 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  A  =  B   &    |-  C  =  B   &    |-  C  =  D   =>    |-  D  =  A
 
Theorem3eqtr3i 2284 An inference from three chained equalities. (Contributed by NM, 6-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  A  =  B   &    |-  A  =  C   &    |-  B  =  D   =>    |-  C  =  D
 
Theorem3eqtr3ri 2285 An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.)
 |-  A  =  B   &    |-  A  =  C   &    |-  B  =  D   =>    |-  D  =  C
 
Theorem3eqtr4i 2286 An inference from three chained equalities. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  A  =  B   &    |-  C  =  A   &    |-  D  =  B   =>    |-  C  =  D
 
Theorem3eqtr4ri 2287 An inference from three chained equalities. (Contributed by NM, 2-Sep-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  A  =  B   &    |-  C  =  A   &    |-  D  =  B   =>    |-  D  =  C
 
Theoremeqtrd 2288 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  =  C )
 
Theoremeqtr2d 2289 An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  C  =  A )
 
Theoremeqtr3d 2290 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =  C )   =>    |-  ( ph  ->  B  =  C )
 
Theoremeqtr4d 2291 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A  =  C )
 
Theorem3eqtrd 2292 A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =  C )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  A  =  D )
 
Theorem3eqtrrd 2293 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =  C )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  D  =  A )
 
Theorem3eqtr2d 2294 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  A  =  D )
 
Theorem3eqtr2rd 2295 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  D  =  A )
 
Theorem3eqtr3d 2296 A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  C  =  D )
 
Theorem3eqtr3rd 2297 A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  D  =  C )
 
Theorem3eqtr4d 2298 A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  A )   &    |-  ( ph  ->  D  =  B )   =>    |-  ( ph  ->  C  =  D )
 
Theorem3eqtr4rd 2299 A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  A )   &    |-  ( ph  ->  D  =  B )   =>    |-  ( ph  ->  D  =  C )
 
Theoremsyl5eq 2300 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  =  C )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31284
  Copyright terms: Public domain < Previous  Next >