P. 22 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2101-2200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eu3 2101*	An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
Theorem	eu5 2102	Uniqueness in terms of "at most one". (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)
⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
Theorem	exmoeu2 2103	Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)
⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))
Theorem	moabs 2104	Absorption of existence condition by "at most one". (Contributed by NM, 4-Nov-2002.)
⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
Theorem	exmodc 2105	If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.)
⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ∨ ∃*𝑥𝜑))
Theorem	exmonim 2106	There is at most one of something which does not exist. Unlike exmodc 2105 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.)
⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
Theorem	mo2r 2107*	A condition which implies "at most one". (Contributed by Jim Kingdon, 2-Jul-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑)
Theorem	mo3h 2108*	Alternate definition of "at most one". Definition of [BellMachover] p. 460, except that definition has the side condition that 𝑦 not occur in 𝜑 in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Theorem	mo3 2109*	Alternate definition of "at most one". Definition of [BellMachover] p. 460, except that definition has the side condition that 𝑦 not occur in 𝜑 in place of our hypothesis. (Contributed by NM, 8-Mar-1995.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Theorem	mo2dc 2110*	Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (DECID ∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
Theorem	euan 2111	Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Theorem	euanv 2112*	Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.)
⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))
Theorem	euor2 2113	Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓))
Theorem	sbmo 2114*	Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)
Theorem	mo4f 2115*	"At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
Theorem	mo4 2116*	"At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
Theorem	eu4 2117*	Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
Theorem	exmoeudc 2118	Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.)
⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑)))
Theorem	moim 2119	"At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
Theorem	moimi 2120	"At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃*𝑥𝜓 → ∃*𝑥𝜑)
Theorem	moimv 2121*	Move antecedent outside of "at most one". (Contributed by NM, 28-Jul-1995.)
⊢ (∃*𝑥(𝜑 → 𝜓) → (𝜑 → ∃*𝑥𝜓))
Theorem	euimmo 2122	Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))
Theorem	euim 2123	Add existential unique existential 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.)
⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑))
Theorem	moan 2124	"At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)
⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜓 ∧ 𝜑))
Theorem	moani 2125	"At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)
⊢ ∃*𝑥𝜑    ⇒   ⊢ ∃*𝑥(𝜓 ∧ 𝜑)
Theorem	moor 2126	"At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)
⊢ (∃*𝑥(𝜑 ∨ 𝜓) → ∃*𝑥𝜑)
Theorem	mooran1 2127	"At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ((∃*𝑥𝜑 ∨ ∃*𝑥𝜓) → ∃*𝑥(𝜑 ∧ 𝜓))
Theorem	mooran2 2128	"At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ (∃*𝑥(𝜑 ∨ 𝜓) → (∃*𝑥𝜑 ∧ ∃*𝑥𝜓))
Theorem	moanim 2129	Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 3-Dec-2001.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
Theorem	moanimv 2130*	Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 23-Mar-1995.)
⊢ (∃*𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
Theorem	moaneu 2131	Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.)
⊢ ∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)
Theorem	moanmo 2132	Nested at-most-one quantifiers. (Contributed by NM, 25-Jan-2006.)
⊢ ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)
Theorem	mopick 2133	"At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.)
⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
Theorem	eupick 2134	Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing 𝑥 such that 𝜑 is true, and there is also an 𝑥 (actually the same one) such that 𝜑 and 𝜓 are both true, then 𝜑 implies 𝜓 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.)
⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
Theorem	eupicka 2135	Version of eupick 2134 with closed formulas. (Contributed by NM, 6-Sep-2008.)
⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓))
Theorem	eupickb 2136	Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)
⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 ↔ 𝜓))
Theorem	eupickbi 2137	Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))
Theorem	mopick2 2138	"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 1655. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	moexexdc 2139	"At most one" double quantification. (Contributed by Jim Kingdon, 5-Jul-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (DECID ∃𝑥𝜑 → ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓)))
Theorem	euexex 2140	Existential uniqueness and "at most one" double quantification. (Contributed by Jim Kingdon, 28-Dec-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ ((∃!𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))
Theorem	2moex 2141	Double quantification with "at most one". (Contributed by NM, 3-Dec-2001.)
⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥𝜑)
Theorem	2euex 2142	Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑)
Theorem	2eumo 2143	Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.)
⊢ (∃!𝑥∃*𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
Theorem	2eu2ex 2144	Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
⊢ (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃𝑦𝜑)
Theorem	2moswapdc 2145	A condition allowing swap of "at most one" and existential quantifiers. (Contributed by Jim Kingdon, 6-Jul-2018.)
⊢ (DECID ∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)))
Theorem	2euswapdc 2146	A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Jim Kingdon, 7-Jul-2018.)
⊢ (DECID ∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑)))
Theorem	2exeu 2147	Double existential uniqueness implies double unique existential quantification. (Contributed by NM, 3-Dec-2001.)
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
Theorem	2eu4 2148*	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 ∃!𝑥∃!𝑦𝜑. See 2exeu 2147 for a one-way implication. (Contributed by NM, 3-Dec-2001.)
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
Theorem	2eu7 2149	Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))
Theorem	euequ1 2150*	Equality has existential uniqueness. (Contributed by Stefan Allan, 4-Dec-2008.)
⊢ ∃!𝑥 𝑥 = 𝑦
Theorem	exists1 2151*	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. (Contributed by NM, 5-Apr-2004.)
⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
Theorem	exists2 2152	A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ ((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)
1.4.7  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 1443) 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 𝜑, 𝜓, and 𝜒 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, https://plato.stanford.edu/entries/quantification/ 1443. See Principia Mathematica page 22 and *10 for more information (especially *10.3 and *10.26). Expressions of the form "no 𝜑 is 𝜓 " are consistently translated as ∀𝑥(𝜑 → ¬ 𝜓). These can also be expressed as ¬ ∃𝑥(𝜑 ∧ 𝜓), per alinexa 1627. We translate "all 𝜑 is 𝜓 " to ∀𝑥(𝜑 → 𝜓), "some 𝜑 is 𝜓 " to ∃𝑥(𝜑 ∧ 𝜓), and "some 𝜑 is not 𝜓 " to ∃𝑥(𝜑 ∧ ¬ 𝜓). 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 ∈ 𝐴, or 𝑥 ⊆ 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 ∈ 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 nonexistent 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 nonexistent 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), https://iep.utm.edu/aristotle-log/ 1627. 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 2157, celaront 2158, cesaro 2163, camestros 2164, felapton 2169, darapti 2170, calemos 2174, fesapo 2175, and bamalip 2176. 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.
Theorem	barbara 2153	"Barbara", one of the fundamental syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and all 𝜒 is 𝜑, therefore all 𝜒 is 𝜓. (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 ∀𝑥(𝑥 ∈ 𝐻 → 𝑥 ∈ 𝑀) (all men are mortal) and ∀𝑥(𝑥 = 𝑆 → 𝑥 ∈ 𝐻) (Socrates is a man) therefore ∀𝑥(𝑥 = 𝑆 → 𝑥 ∈ 𝑀) (Socrates is mortal). Russell and Whitehead note that the "syllogism in Barbara is derived..." from syl 14. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1659. There are a legion of sources for Barbara, including https://www.friesian.com/aristotl.htm 1659, https://plato.stanford.edu/entries/aristotle-logic/ 1659, and https://en.wikipedia.org/wiki/Syllogism 1659. (Contributed by David A. Wheeler, 24-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    ⇒   ⊢ ∀𝑥(𝜒 → 𝜓)
Theorem	celarent 2154	"Celarent", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and all 𝜒 is 𝜑, therefore no 𝜒 is 𝜓. (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.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    ⇒   ⊢ ∀𝑥(𝜒 → ¬ 𝜓)
Theorem	darii 2155	"Darii", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜒 is 𝜑, therefore some 𝜒 is 𝜓. (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.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ 𝜑)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	ferio 2156	"Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜑, therefore some 𝜒 is not 𝜓. (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.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ 𝜑)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	barbari 2157	"Barbari", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is 𝜓. (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.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	celaront 2158	"Celaront", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is not 𝜓. (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.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	cesare 2159	"Cesare", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and all 𝜒 is 𝜓, therefore no 𝜒 is 𝜑. (In Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to celarent 2154. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜓)    ⇒   ⊢ ∀𝑥(𝜒 → ¬ 𝜑)
Theorem	camestres 2160	"Camestres", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and no 𝜒 is 𝜓, therefore no 𝜒 is 𝜑. (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.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → ¬ 𝜓)    ⇒   ⊢ ∀𝑥(𝜒 → ¬ 𝜑)
Theorem	festino 2161	"Festino", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜓, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ 𝜓)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	baroco 2162	"Baroco", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜒 is not 𝜓, therefore some 𝜒 is not 𝜑. (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.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	cesaro 2163	"Cesaro", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜒 is 𝜓, and 𝜒 exist, therefore some 𝜒 is not 𝜑. (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.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜓)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	camestros 2164	"Camestros", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, no 𝜒 is 𝜓, and 𝜒 exist, therefore some 𝜒 is not 𝜑. (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.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → ¬ 𝜓)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	datisi 2165	"Datisi", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∃𝑥(𝜑 ∧ 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	disamis 2166	"Disamis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. (In Aristotelian notation, IAI-3: MiP and MaS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∃𝑥(𝜑 ∧ 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	ferison 2167	"Ferison", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is not 𝜓. (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.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜑 ∧ 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	bocardo 2168	"Bocardo", one of the syllogisms of Aristotelian logic. Some 𝜑 is not 𝜓, and all 𝜑 is 𝜒, therefore some 𝜒 is not 𝜓. (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 2166; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.)
⊢ ∃𝑥(𝜑 ∧ ¬ 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	felapton 2169	"Felapton", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is not 𝜓. (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.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	darapti 2170	"Darapti", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is 𝜓. (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.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	calemes 2171	"Calemes", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and no 𝜓 is 𝜒, therefore no 𝜒 is 𝜑. (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.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜓 → ¬ 𝜒)    ⇒   ⊢ ∀𝑥(𝜒 → ¬ 𝜑)
Theorem	dimatis 2172	"Dimatis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜓 is 𝜒, therefore some 𝜒 is 𝜑. (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 2155 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∃𝑥(𝜑 ∧ 𝜓)    &   ⊢ ∀𝑥(𝜓 → 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜑)
Theorem	fresison 2173	"Fresison", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓 (PeM), and some 𝜓 is 𝜒 (MiS), therefore some 𝜒 is not 𝜑 (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.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜓 ∧ 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	calemos 2174	"Calemos", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓 (PaM), no 𝜓 is 𝜒 (MeS), and 𝜒 exist, therefore some 𝜒 is not 𝜑 (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.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜓 → ¬ 𝜒)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	fesapo 2175	"Fesapo", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜓 is 𝜒, and 𝜓 exist, therefore some 𝜒 is not 𝜑. (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.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜓 → 𝜒)    &   ⊢ ∃𝑥𝜓    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	bamalip 2176	"Bamalip", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜓 is 𝜒, and 𝜑 exist, therefore some 𝜒 is 𝜑. (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 2157. (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜓 → 𝜒)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜑)
1.5  First-order logic with one non-logical binary predicate This section adds one non-logical binary predicate to the first-order logic developed until here. We call it "the membership predicate" since it will be used in the next part as the membership predicate of set theory, but in this section, it has no other property than being "a binary predicate". "Non-logical" means that it does not belong to the logic. In our logic (and in most treatments), the only logical predicate is the equality predicate (see weq 1527).
Syntax	wcel 2177	Extend wff definition to include the membership connective between classes. For a general discussion of the theory of classes, see mmset.html#class. The purpose of introducing wff 𝐴 ∈ 𝐵 here is to allow us to prove the wel 2178 of predicate calculus in terms of the wceq 1373 of set theory, so that we do not overload the ∈ connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables 𝐴 and 𝐵 are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-clab 2193 for more information on the set theory usage of wcel 2177.
wff 𝐴 ∈ 𝐵
Theorem	wel 2178	Extend wff definition to include atomic formulas with the membership predicate. This is read either "𝑥 is an element of 𝑦", or "𝑥 is a member of 𝑦", or "𝑥 belongs to 𝑦", or "𝑦 contains 𝑥". Note: The phrase "𝑦 includes 𝑥 " means "𝑥 is a subset of 𝑦"; to use it also for 𝑥 ∈ 𝑦, as some authors occasionally do, is poor form and causes confusion, according to George Boolos (1992 lecture at MIT). This syntactical construction introduces a binary non-logical predicate symbol ∈ into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for ∈ apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments. Instead of introducing wel 2178 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wcel 2177. This lets us avoid overloading the ∈ connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 2178 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 2177. Note: To see the proof steps of this syntax proof, type "MM> SHOW PROOF wel / ALL" in the Metamath program. (Contributed by NM, 24-Jan-2006.)
wff 𝑥 ∈ 𝑦
Axiom	ax-13 2179	Axiom of left equality for the binary predicate ∈. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of the ∈ binary predicate. Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
Axiom	ax-14 2180	Axiom of right equality for the binary predicate ∈. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of the ∈ binary predicate. Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
Theorem	elequ1 2181	An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
Theorem	elequ2 2182	An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
Theorem	cleljust 2183*	When the class variables of set theory are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 2178 with the class variables in wcel 2177. (Contributed by NM, 28-Jan-2004.)
⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦))
Theorem	elsb1 2184*	Substitution for the first argument of the non-logical predicate in an atomic formula. See elsb2 2185 for substitution for the second argument. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)
Theorem	elsb2 2185*	Substitution for the second argument of the non-logical predicate in an atomic formula. See elsb1 2184 for substitution for the first argument. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)
Theorem	dveel1 2186*	Quantifier introduction when one pair of variables is disjoint. (Contributed by NM, 2-Jan-2002.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 ∈ 𝑧 → ∀𝑥 𝑦 ∈ 𝑧))
Theorem	dveel2 2187*	Quantifier introduction when one pair of variables is disjoint. (Contributed by NM, 2-Jan-2002.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → ∀𝑥 𝑧 ∈ 𝑦))
PART 2  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 an element of another set, and this relationship is indicated by the ∈ 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. Here we develop set theory based on the Intuitionistic Zermelo-Fraenkel (IZF) system, mostly following the IZF axioms as laid out in [Crosilla]. Constructive Zermelo-Fraenkel (CZF), also described in Crosilla, is not as easy to formalize in Metamath because the statement of some of its axioms uses the notion of "bounded formula". Since Metamath has, purposefully, a very weak metalogic, that notion must be developed in the logic itself. This is similar to our treatment of substitution (df-sb 1787) and our definition of the nonfreeness predicate (df-nf 1485), whereas substitution and bound and free variables are ordinarily defined in the metalogic. The development of CZF has begun in BJ's mathbox, see wbd 15822.
2.1  IZF Set Theory - start with the Axiom of Extensionality
2.1.1  Introduce the Axiom of Extensionality
Axiom	ax-ext 2188*	Axiom of Extensionality. It states that two sets are identical if they contain the same elements. Axiom 1 of [Crosilla] p. "Axioms of CZF and IZF" (with unnecessary quantifiers removed). Set theory can also be formulated with a single primitive predicate ∈ on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes (∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)), and equality 𝑥 = 𝑦 is defined as ∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 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 1528 through ax-16 1838 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. 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 2188 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 axioms which involve wff metavariables). 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 2188 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
Theorem	axext3 2189*	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.)
⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
Theorem	axext4 2190*	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 2188. (Contributed by NM, 14-Nov-2008.)
⊢ (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
Theorem	bm1.1 2191*	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.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) → ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑))
2.1.2  Class abstractions (a.k.a. class builders)
Syntax	cab 2192	Introduce the class builder or class abstraction notation ("the class of sets 𝑥 such that 𝜑 is true"). Our class variables 𝐴, 𝐵, etc. range over class builders (sometimes implicitly). Note that a setvar variable can be expressed as a class builder per Theorem cvjust 2201, justifying the assignment of setvar variables to class variables via the use of cv 1372.
class {𝑥 ∣ 𝜑}
Definition	df-clab 2193	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, 𝜑 will have 𝑦 as a free variable, and "{𝑦 ∣ 𝜑} " is read "the class of all sets 𝑦 such that 𝜑(𝑦) is true". We do not define {𝑦 ∣ 𝜑} in isolation but only as part of an expression that extends or "overloads" the ∈ relationship. This is our first use of the ∈ symbol to connect classes instead of sets. The syntax definition wcel 2177, which extends or "overloads" the wel 2178 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2199 and df-clel 2202, we introduce a new kind of variable (class variable) that can substituted with expressions such as {𝑦 ∣ 𝜑}. In the present definition, the 𝑥 on the left-hand side is a setvar variable. Syntax definition cv 1372 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2201 (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 2315 for a quick overview). Because class variables can be substituted with compound expressions and setvar variables cannot, it is often useful to convert a theorem containing a free setvar variable to a more general version with a class variable. 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 ∣ 𝜑} a "class term". For a general discussion of the theory of classes, see https://us.metamath.org/mpeuni/mmset.html#class 2315. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
Theorem	abid 2194	Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
Theorem	hbab1 2195*	Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})
Theorem	nfsab1 2196*	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
Theorem	hbab 2197*	Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑})
Theorem	nfsab 2198*	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}
Definition	df-cleq 2199*	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 ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧), which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see Theorem axext4 2190). 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 2193, df-clel 2202, and abeq2 2315. In the form of dfcleq 2200, 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 https://us.metamath.org/mpeuni/mmset.html#class 2200. (Contributed by NM, 15-Sep-1993.)
⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧)    ⇒   ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	dfcleq 2200*	The same as df-cleq 2199 with the hypothesis removed using the Axiom of Extensionality ax-ext 2188. (Contributed by NM, 15-Sep-1993.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))