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Theorem List for Metamath Proof Explorer - 1601-1700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem2albiim 1601 Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.)

Theoremexintrbi 1602 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)

Theoremexintr 1603 Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)

Theoremalsyl 1604 Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)

1.4.4  Axiom scheme ax-17 (Distinctness) - first use of \$d

Axiomax-17 1605* Axiom of Distinctness. This axiom quantifies a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113.

(See comments in ax17o 2098 about the logical redundancy of ax-17 1605 in the presence of our obsolete axioms.)

This axiom essentially says that if does not occur in , i.e. does not depend on in any way, then we can add the quantifier to with no further assumptions. By sp 1718, we can also remove the quantifier (unconditionally). (Contributed by NM, 5-Aug-1993.)

Theorema17d 1606* ax-17 1605 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.)

Theoremnfv 1607* If is not present in , then is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfvd 1608* nfv 1607 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1763. (Contributed by Mario Carneiro, 6-Oct-2016.)

Theoremalimdv 1609* Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.)

Theoremeximdv 1610* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)

Theorem2alimdv 1611* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.)

Theorem2eximdv 1612* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.)

Theoremalbidv 1613* Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)

Theoremexbidv 1614* Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)

Theorem2albidv 1615* Formula-building rule for 2 universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)

Theorem2exbidv 1616* Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)

Theorem3exbidv 1617* Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)

Theorem4exbidv 1618* Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)

Theoremalrimiv 1619* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theoremalrimivv 1620* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)

Theoremalrimdv 1621* Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.)

Theoremnfdv 1622* Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theorem2ax17 1623* Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)

1.4.5  Equality predicate; define substitution

Syntaxcv 1624 This syntax construction states that a variable , which has been declared to be a set variable by \$f statement vx, is also a class expression. This can be justified informally as follows. We know that the class builder is a class by cab 2271. Since (when is distinct from ) we have by cvjust 2280, we can argue that the syntax " " can be viewed as an abbreviation for " ". See the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class."

While it is tempting and perhaps occasionally useful to view cv 1624 as a "type conversion" from a set variable to a class variable, keep in mind that cv 1624 is intrinsically no different from any other class-building syntax such as cab 2271, cun 3152, or c0 3457.

For a general discussion of the theory of classes and the role of cv 1624, see http://us.metamath.org/mpeuni/mmset.html#class.

(The description above applies to set theory, not predicate calculus. The purpose of introducing here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1626 of predicate calculus from the wceq 1625 of set theory, so that we don't "overload" the connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.)

Syntaxwceq 1625 Extend wff definition to include class equality.

For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class.

(The purpose of introducing here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1626 of predicate calculus in terms of the wceq 1625 of set theory, so that we don't "overload" the connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. For example, some parsers - although not the Metamath program - stumble on the fact that the in could be the of either weq 1626 or wceq 1625, although mathematically it makes no difference. The class variables and are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 2278 for more information on the set theory usage of wceq 1625.)

Theoremweq 1626 Extend wff definition to include atomic formulas using the equality predicate.

(Instead of introducing weq 1626 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 wceq 1625. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1626 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1625. Note: To see the proof steps of this syntax proof, type "show proof weq /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

Theoremequs3 1627 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)

Theoremspeimfw 1628 Specialization, with additional weakening to allow bundling of and . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Aug-2017.)

Theoremspimfw 1629 Specialization, with additional weakening to allow bundling of and . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)

Theoremax11i 1630 Inference that has ax-11 1717 (without ) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)

Syntaxwsb 1631 Extend wff definition to include proper substitution (read "the wff that results when is properly substituted for in wff "). (Contributed by NM, 24-Jan-2006.)

Definitiondf-sb 1632 Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use to mean "the wff that results from the proper substitution of for in the wff ." We can also use in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 1966.

Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, " is the wff that results when is properly substituted for in ." For example, if the original is , then is , from which we obtain that is . So what exactly does mean? Curry's notation solves this problem.

In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 2002, sbcom2 2055 and sbid2v 2064).

Note that our definition is valid even when and are replaced with the same variable, as sbid 1865 shows. We achieve this by having free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 2060 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another version that mixes free and bound variables is dfsb3 1998. When and are distinct, we can express proper substitution with the simpler expressions of sb5 2041 and sb6 2040.

There are no restrictions on any of the variables, including what variables may occur in wff . (Contributed by NM, 5-Aug-1993.)

Theoremsbequ2 1633 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsb1 1634 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbimi 1635 Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)

Theoremsbbii 1636 Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)

1.4.6  Axiom scheme ax-9 (Existence)

Axiomax-9 1637 Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. This axiom tells us is that at least one thing exists. In this form (not requiring that and be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) It is equivalent to axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint); the equivalence is established by ax9o 1892 and ax9from9o 2089. A more convenient form of this axiom is a9e 1893, which has additional remarks.

Raph Levien proved the independence of this axiom from the other logical axioms on 12-Apr-2005. See item 16 at http://us.metamath.org/award2003.html.

ax-9 1637 can be proved from the weaker version ax9v 1638 requiring that the variables be distinct; see theorem ax9 1891.

ax-9 1637 can also be proved from the Axiom of Separation (in the form that we use that axiom, where free variables are not universally quantified). See theorem ax9vsep 4147.

Except by ax9v 1638, this axiom should not be referenced directly. Instead, use theorem ax9 1891. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremax9v 1638* Axiom B7 of [Tarski] p. 75, which requires that and be distinct. This trivial proof is intended merely to weaken axiom ax-9 1637 by adding a distinct variable restriction. From here on, ax-9 1637 should not be referenced directly by any other proof, so that theorem ax9 1891 will show that we can recover ax-9 1637 from this weaker version if it were an axiom (as it is in the case of Tarski).

Note: Introducing as a distinct variable group "out of the blue" with no apparent justification has puzzled some people, but it is perfectly sound. All we are doing is adding an additional redundant requirement, no different from adding a redundant logical hypothesis, that results in a weakening of the theorem. This means that any future theorem that references ax9v 1638 must have a \$d specified for the two variables that get substituted for and . The \$d does not propagate "backwards" i.e. it does not impose a requirement on ax-9 1637.

When possible, use of this theorem rather than ax9 1891 is preferred since its derivation from axioms is much shorter. (Contributed by NM, 7-Aug-2015.)

Theorema9ev 1639* At least one individual exists. Weaker version of a9e 1893. When possible, use of this theorem rather than a9e 1893 is preferred since its derivation from axioms is much shorter. (Contributed by NM, 3-Aug-2017.)

Theoremspimw 1640* Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)

Theoremspimvw 1641* Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Theoremspnfw 1642 Weak version of sp 1718. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 13-Aug-2017.)

Theoremcbvaliw 1643* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.)

Theoremcbvalivw 1644* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)

1.4.7  Axiom scheme ax-8 (Equality)

Axiomax-8 1645 Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. This is similar to, but not quite, a transitive law for equality (proved later as equtr 1654). This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom C7 of [Monk2] p. 105 and Axiom Scheme C8' in [Megill] p. 448 (p. 16 of the preprint).

The equality symbol was invented in 1527 by Robert Recorde. He chose a pair of parallel lines of the same length because "noe .2. thynges, can be moare equalle."

Note that this axiom is still valid even when any two or all three of , , and are replaced with the same variable since they do not have any distinct variable (Metamath's \$d) restrictions. Because of this, we say that these three variables are "bundled" (a term coined by Raph Levien). (Contributed by NM, 5-Aug-1993.)

Theoremequid 1646 Identity law for equality. Lemma 2 of [KalishMontague] p. 85. See also Lemma 6 of [Tarski] p. 68. (Contributed by NM, 1-Apr-2005.) (Revised by NM, 9-Apr-2017.)

Theoremnfequid 1647 Bound-variable hypothesis builder for . This theorem tells us that any variable, including , is effectively not free in , even though is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)

Theoremequcomi 1648 Commutative law for equality. Lemma 3 of [KalishMontague] p. 85. See also Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 9-Apr-2017.)

Theoremequcom 1649 Commutative law for equality. (Contributed by NM, 20-Aug-1993.)

Theoremequequ1 1650 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)

Theoremequequ2 1651 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.)

Theoremstdpc6 1652 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1860.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)

Theoremequcoms 1653 An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.)

Theoremequtr 1654 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)

Theoremequtrr 1655 A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)

Theoremequtr2 1656 A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremax12b 1657 Two equivalent ways of expressing ax-12 1868. See the comment for ax-12 1868. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 12-Aug-2017.)

Theoremax12bOLD 1658 Obsolete version of ax12b 1657 as of 12-Aug-2017. (Contributed by NM, 2-May-2017.) (New usage is discouraged.)

Theoremspfw 1659* Weak version of sp 1718. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. TO DO: Do we need this theorem? If not, maybe it should be deleted. (Contributed by NM, 19-Apr-2017.)

TheoremspnfwOLD 1660 Weak version of sp 1718. Uses only Tarski's FOL axiom schemes. Obsolete version of spnfw 1642 as of 13-Aug-2017. (Contributed by NM, 1-Aug-2017.) (New usage is discouraged.)

Theorem19.8w 1661 Weak version of 19.8a 1720. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.)

Theoremspw 1662* Weak version of specialization scheme sp 1718. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 1718 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 1718 having no wff metavariables and mutually distinct set variables (see ax11wdemo 1699 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 1718 are spfw 1659 (minimal distinct variable requirements), spnfw 1642 (when is not free in ), spvw 1663 (when does not appear in ), sptruw 1671 (when is true), and spfalw 1672 (when is false). (Contributed by NM, 9-Apr-2017.)

Theoremspvw 1663* Version of sp 1718 when does not occur in . This provides the other direction of ax-17 1605. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.)

Theorem19.3v 1664* Special case of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 1-Aug-2017.)

Theorem19.9v 1665* Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.) (Revised by NM, 1-Aug-2017.)

Theoremexlimdv 1666* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)

Theoremexlimddv 1667* Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)

Theoremexlimiv 1668* Inference from Theorem 19.23 of [Margaris] p. 90.

This inference, along with our many variants such as rexlimdv 2668, is used to implement a metatheorem called "Rule C" that is given in many logic textbooks. See, for example, Rule C in [Mendelson] p. 81, Rule C in [Margaris] p. 40, or Rule C in Hirst and Hirst's A Primer for Logic and Proof p. 59 (PDF p. 65) at http://www.mathsci.appstate.edu/~hirstjl/primer/hirst.pdf.

In informal proofs, the statement "Let be an element such that..." almost always means an implicit application of Rule C.

In essence, Rule C states that if we can prove that some element exists satisfying a wff, i.e. where has free, then we can use as a hypothesis for the proof where is a new (ficticious) constant not appearing previously in the proof, nor in any axioms used, nor in the theorem to be proved. The purpose of Rule C is to get rid of the existential quantifier.

We cannot do this in Metamath directly. Instead, we use the original (containing ) as an antecedent for the main part of the proof. We eventually arrive at where is the theorem to be proved and does not contain . Then we apply exlimiv 1668 to arrive at . Finally, we separately prove and detach it with modus ponens ax-mp 8 to arrive at the final theorem . (Contributed by NM, 5-Aug-1993.)

Theoremexlimivv 1669* Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.)

Theoremexlimdvv 1670* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)

Theoremsptruw 1671 Version of sp 1718 when is true. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.)

Theoremspfalw 1672 Version of sp 1718 when is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.)

Theorem19.2 1673 Theorem 19.2 of [Margaris] p. 89. Note: This proof is very different from Margaris' because we only have Tarski's FOL axiom schemes available at this point. See the later 19.2g 1782 for a more conventional proof. (Contributed by NM, 2-Aug-2017.)

Theorem19.39 1674 Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.24 1675 Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.34 1676 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theoremcbvalw 1677* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Theoremcbvalvw 1678* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Theoremcbvexvw 1679* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)

Theoremalcomiw 1680* Weak version of alcom 1713. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.)

Theoremhbn1fw 1681* Weak version of ax-6 1705 from which we can prove any ax-6 1705 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)

Theoremhbn1w 1682* Weak version of hbn1 1706. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Theoremhba1w 1683* Weak version of hba1 1721. See comments for ax6w 1693. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Theoremhbe1w 1684* Weak version of hbe1 1707. See comments for ax6w 1693. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)

Theoremhbalw 1685* Weak version of hbal 1712. Uses only Tarski's FOL axiom schemes. Unlike hbal 1712, this theorem requires that and be distinct i.e. are not bundled. (Contributed by NM, 19-Apr-2017.)

1.4.8  Membership predicate

Syntaxwcel 1686 Extend wff definition to include the membership connective between classes.

For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class.

(The purpose of introducing here is to allow us to express i.e. "prove" the wel 1687 of predicate calculus in terms of the wceq 1625 of set theory, so that we don't "overload" the connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables and are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-clab 2272 for more information on the set theory usage of wcel 1686.)

Theoremwel 1687 Extend wff definition to include atomic formulas with the epsilon (membership) predicate. This is read " is an element of ," " is a member of ," " belongs to ," or " contains ." Note: The phrase " includes " means " is a subset of ;" to use it also for , 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 (epsilon) 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 1687 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 1686. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1687 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1686. Note: To see the proof steps of this syntax proof, type "show proof wel /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

1.4.9  Axiom schemes ax-13 (Left Membership Equality)

Axiomax-13 1688 Axiom of Left Membership Equality. 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. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). "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.)

Theoremelequ1 1689 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)

1.4.10  Axiom schemes ax-14 (Right Membership Equality)

Axiomax-14 1690 Axiom of Right Membership Equality. 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. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)

Theoremelequ2 1691 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)

1.4.11  Logical redundancy of ax-6 , ax-7 , ax-11 , ax-12

The orginal axiom schemes of Tarski's predicate calculus are ax-5 1546, ax-17 1605, ax9v 1638, ax-8 1645, ax-13 1688, and ax-14 1690, together with rule ax-gen 1535. See http://us.metamath.org/mpeuni/mmset.html#compare 1535. They are given as axiom schemes B4 through B8 in [KalishMontague] p. 81. These are shown to be logically complete by Theorem 1 of [KalishMontague] p. 85.

The axiom system of set.mm includes the auxiliary axiom schemes ax-6 1705, ax-7 1710, ax-12 1868, and ax-11 1717, which are not part of Tarski's axiom schemes. They are used (and we conjecture are required) to make our system "metalogically complete" i.e. able to prove directly all possible schemes with wff and set metavariables, bundled or not, whose object-language instances are valid. (ax-11 1717 has been proved to be required; see http://us.metamath.org/award2003.html#9a. Metalogical independence of the other three are open problems.)

(There are additional predicate calculus axiom schemes included in set.mm such as ax-4 2076, but they can all be proved as theorems from the above.)

Terminology: Two set (individual) metavariables are "bundled" in an axiom or theorem scheme when there is no distinct variable constraint (\$d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the and in ax9 1891 are bundled, but they are not in ax9v 1638. We also say that a scheme is bundled when it has at least one pair of bundled set metavariables. If distinct variable conditions are added to all set metavariable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax9v 1638 is the principal instance of ax9 1891. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance of ax9 1891 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions (\$d) to be concerned with. There is also a small economy in being able to state principal and degenerate instances simultaneously. A disadvantage is that bundling may present difficulties in translations to other proof languages, which typically lack the concept (in part because their variables often represent the variables of the object language rather than metavariables ranging over them).

Because Tarski's axiom schemes are logically complete, they can be used to prove any object-language instance of ax-6 1705, ax-7 1710, ax-11 1717, and ax-12 1868 . "Translating" this to Metamath, it means that Tarski's axioms can prove any substitution instance of ax-6 1705, ax-7 1710, ax-11 1717, or ax-12 1868 in which (1) there are no wff metavariables and (2) all set metavariables are mutually distinct i.e. are not bundled. In effect this is mimicking the object language by pretending that each set metavariable is an object-language variable. (There may also be specific instances with wff metavariables and/or bundling that are directly provable from Tarski's axiom schemes, but it isn't guaranteed. Whether all of them are possible is part of the still open metalogical independence problem for our additional axiom schemes.)

It can be useful to see how this can be done, both to show that our additional schemes are valid metatheorems of Tarski's system and to be able to translate object language instances of our proofs into proofs that would work with a system using only Tarski's original schemes. In addition, it may (or may not) provide insight into the conjectured metalogical independence of our additional schemes.

The new theorem schemes ax6w 1693, ax7w 1694, ax11w 1697, and ax12w 1700 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-6 1705, ax-7 1710, ax-11 1717, and ax-12 1868 meeting conditions (1) and (2). (The "w" suffix stands for "weak version".) Each hypothesis of ax6w 1693, ax7w 1694, and ax11w 1697 is of the form where is an auxiliary or "dummy" wff metavariable in which doesn't occur. We can show by induction on formula length that the hypotheses can be eliminated in all cases meeting conditions (1) and (2). The example ax11wdemo 1699 illustrates the techniques (equality theorems and bound variable renaming) used to achieve this.

We also show the degenerate instances for axioms with bundled variables in ax7dgen 1695, ax11dgen 1698, ax12dgen1 1701, ax12dgen2 1702, ax12dgen3 1703, and ax12dgen4 1704. (Their proofs are trivial, but we include them to be thorough.) Combining the principal and degenerate cases outside of Metamath, we show that the bundled schemes ax-6 1705, ax-7 1710, ax-11 1717, and ax-12 1868 are schemes of Tarski's system, meaning that all object language instances they generate are theorems of Tarski's system.

It is interesting that Tarski used the bundled scheme ax-9 1637 in an older system, so it seems the main purpose of his later ax9v 1638 was just to show that the weaker unbundled form is sufficient rather than an aesthetic objection to bundled free and bound variables. Since we adopt the bundled ax-9 1637 as our official axiom, we show that the degenerate instance holds in ax9dgen 1692.

The case of sp 1718 is curious: originally an axiom of Tarski's system, it was proved redundant by Lemma 9 of [KalishMontague] p. 86. However, the proof is by induction on formula length, and the compact scheme form apparently cannot be proved directly from Tarski's other axioms. The best we can do seems to be spw 1662, again requiring substitution instances of that meet conditions (1) and (2) above. Note that our direct proof sp 1718 requires ax-11 1717, which is not part of Tarski's system.

Theoremax9dgen 1692 Tarski's system uses the weaker ax9v 1638 instead of the bundled ax-9 1637, so here we show that the degenerate case of ax-9 1637 can be derived. (Contributed by NM, 23-Apr-2017.)

Theoremax6w 1693* Weak version of ax-6 1705 from which we can prove any ax-6 1705 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Theoremax7w 1694* Weak version of ax-7 1710 from which we can prove any ax-7 1710 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-7 1710, this theorem requires that and be distinct i.e. are not bundled. (Contributed by NM, 10-Apr-2017.)

Theoremax7dgen 1695 Degenerate instance of ax-7 1710 where bundled variables and have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)

Theoremax11wlem 1696* Lemma for weak version of ax-11 1717. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax11w 1697. (Contributed by NM, 10-Apr-2017.)

Theoremax11w 1697* Weak version of ax-11 1717 from which we can prove any ax-11 1717 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. An instance of the first hypothesis will normally require that and be distinct (unless does not occur in ). (Contributed by NM, 10-Apr-2017.)

Theoremax11dgen 1698 Degenerate instance of ax-11 1717 where bundled variables and have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)

Theoremax11wdemo 1699* Example of an application of ax11w 1697 that results in an instance of ax-11 1717 for a contrived formula with mixed free and bound variables, , in place of . The proof illustrates bound variable renaming with cbvalvw 1678 to obtain fresh variables to avoid distinct variable clashes. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 14-Apr-2017.)

Theoremax12w 1700* Weak version (principal instance) of ax-12 1868 not involving bundling. Uses only Tarski's FOL axiom schemes. The proof is trivial but is included to complete the set ax6w 1693, ax7w 1694, and ax11w 1697. (Contributed by NM, 10-Apr-2017.)

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