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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | chvar 1601 | Implicit substitution of for into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) |
Theorem | equvini 1602 | A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require to be distinct from and (making the proof longer). (The proof was shortened by Andrew Salmon, 25-May-2011.) |
Theorem | equveli 1603 | A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1602.) |
Syntax | wsbc 1604 |
Extend wff notation to include the proper substitution of a class for a
set. Read this notation as "the proper substitution of class for
set variable
in wff ."
(The purpose of introducing here is to allow us to express i.e. "prove" the wsb 1605 of predicate calculus in terms of the wsbc 1604 of set theory, so that we don't "overload" its connectives with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variable is introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus.) |
Theorem | wsb 1605 |
Extend wff definition to include proper substitution (read "the wff that
results when is
properly substituted for in wff ").
(Instead of introducing wsb 1605 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 wsbc 1604. This lets us avoid overloading its connectives, thus preventing ambiguity that would complicate some Metamath parsers. Note: To see the proof steps of this syntax proof, type "show proof wsb /all" in the Metamath program.) |
Definition | df-sb 1606 |
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 when
is properly substituted 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 1619.
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 1665, sbcom2 1782 and sbid2v 1790). Note that our definition is valid even when and are replaced with the same variable, as sbid 1618 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 1787 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 1662. When and are distinct, we can express proper substitution with the simpler expressions of sb5 1706 and sb6 1705. There are no restrictions on any of the variables, including what variables may occur in wff . |
Theorem | sbimi 1607 | Infer substitution into antecedent and consequent of an implication. |
Theorem | sbbii 1608 | Infer substitution into both sides of a logical equivalence. |
Theorem | drsb1 1609 | Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). |
Theorem | sb1 1610 | One direction of a simplified definition of substitution. |
Theorem | sb2 1611 | One direction of a simplified definition of substitution. |
Theorem | sbequ1 1612 | An equality theorem for substitution. |
Theorem | sbequ2 1613 | An equality theorem for substitution. |
Theorem | stdpc7 1614 | One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1563.) Translated to traditional notation, it can be read: " , , , provided that is free for in , ." Axiom 7 of [Mendelson] p. 95. |
Theorem | sbequ12 1615 | An equality theorem for substitution. |
Theorem | sbequ12r 1616 | An equality theorem for substitution. (The proof was shortened by Andrew Salmon, 21-Jun-2011.) |
Theorem | sbequ12a 1617 | An equality theorem for substitution. |
Theorem | sbid 1618 | An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). |
Theorem | stdpc4 1619 | The specialization axiom of standard predicate calculus. It states that if a statement holds for all , then it also holds for the specific case of (properly) substituted for . Translated to traditional notation, it can be read: " , provided that is free for in ." Axiom 4 of [Mendelson] p. 69. |
Theorem | sbf 1620 | Substitution for a variable not free in a wff does not affect it. |
Theorem | sbf2 1621 | Substitution has no effect on a bound variable. |
Theorem | sb6x 1622 | Equivalence involving substitution for a variable not free. (The proof was shortened by Andrew Salmon, 12-Aug-2011.) |
Theorem | hbs1f 1623 | If is not free in , it is not free in . (The proof was shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sbequ5 1624 | Substitution does not change an identical variable specifier. |
Theorem | sbequ6 1625 | Substitution does not change a distinctor. |
Theorem | sbt 1626 | A substitution into a theorem remains true. (See chvar 1601 and chvarv 1769 for versions using implicit substitition.) (The proof was shortened by Andrew Salmon, 25-May-2011.) |
Theorem | equsb1 1627 | Substitution applied to an atomic wff. |
Theorem | equsb2 1628 | Substitution applied to an atomic wff. |
Theorem | sbied 1629 | Conversion of implicit substitution to explicit substitution (deduction version of sbie 1630). (The proof was shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sbie 1630 | Conversion of implicit substitution to explicit substitution. |
Theorem | equs5a 1631 | A property related to substitution that unlike equs5 1657 doesn't require a distinctor antecedent. |
Theorem | equs5e 1632 | A property related to substitution that unlike equs5 1657 doesn't require a distinctor antecedent. |
Theorem | sb4a 1633 | A version of sb4 1659 that doesn't require a distinctor antecedent. |
Theorem | equs45f 1634 | Two ways of expressing substitution when is not free in . |
Theorem | sb6f 1635 | Equivalence for substitution when is not free in . |
Theorem | sb5f 1636 | Equivalence for substitution when is not free in . |
Theorem | sb4e 1637 | One direction of a simplified definition of substitution that unlike sb4 1659 doesn't require a distinctor antecedent. |
Theorem | hbsb2a 1638 | Special case of a bound-variable hypothesis builder for substitution. |
Theorem | hbsb2e 1639 | Special case of a bound-variable hypothesis builder for substitution. |
Theorem | hbsb3 1640 | If is not free in , is not free in . |
Theorem | a4imv 1641* | A version of a4im 1592 with a distinct variable requirement instead of a bound variable hypothesis. |
Theorem | aev 1642* | A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1644. (The proof was shortened by Andrew Salmon, 21-Jun-2011.) |
Theorem | ax16 1643* |
Theorem showing that ax-16 1644 is redundant if ax-17 1402 is included in the
axiom system. The important part of the proof is provided by aev 1642.
See ax16ALT 1708 for an alternate proof that does not require ax-10 1388 or ax-12 1393. This theorem should not be referenced in any proof. Instead, use ax-16 1644 below so that theorems needing ax-16 1644 can be more easily identified. |
Axiom | ax-16 1644* |
Axiom of Distinct Variables. The only axiom of predicate calculus
requiring that variables be distinct (if we consider ax-17 1402 to be a
metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p.
16 of the preprint). It apparently does not otherwise appear in the
literature but is easily proved from textbook predicate calculus by
cases. It is a somewhat bizarre axiom since the antecedent is always
false in set theory, but nonetheless it is technically
necessary as you can see from its uses.
This axiom is redundant if we include ax-17 1402; see theorem ax16 1643. Alternately, ax-17 1402 becomes logically redundant in the presence of this axiom, but without ax-17 1402 we lose the more powerful metalogic that results from being able to express the concept of a set variable not occurring in a wff (as opposed to just two set variables being distinct). We retain ax-16 1644 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-17 1402, which might be easier to study for some theoretical purposes. |
Theorem | ax17eq 1645* | Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 1402 considered as a metatheorem. Do not use it for later proofs - use ax-17 1402 instead, to avoid reference to the redundant axiom ax-16 1644.) |
Theorem | dveeq2 1646* | Quantifier introduction when one pair of variables is distinct. |
Theorem | dveeq2ALT 1647* | Version of dveeq2 1646 using ax-16 1644 instead of ax-17 1402. |
Theorem | dvelimfALT2 1648* | Proof of dvelimf 1688 using dveeq2 1646 (shown as the last hypothesis) instead of ax-12 1393. This shows that ax-12 1393 could be replaced by dveeq2 1646 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.) |
Theorem | nd5 1649* | A lemma for proving conditionless ZFC axioms. |
Theorem | exlimdv 1650* | Deduction from Theorem 19.23 of [Margaris] p. 90. |
Theorem | ax11v2 1651* | Recovery of ax11o 1653 from ax11v 1703 without using ax-11 1389. The hypothesis is even weaker than ax11v 1703, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1653. |
Theorem | ax11a2 1652* | Derive ax-11o 1654 from a hypothesis in the form of ax-11 1389. The hypothesis is even weaker than ax-11 1389, with both distinct from and not occurring in . Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1653. As theorem ax11 1655 shows, the distinct variable conditions are optional. An open problem is whether ax11o 1653 can be derived from ax-11 1389 without relying on ax-17 1402. |
Theorem | ax11o 1653 |
Derivation of set.mm's original ax-11o 1654 from the shorter ax-11 1389 that
has replaced it.
An open problem is whether this theorem can be proved without relying on ax-16 1644 or ax-17 1402. Another open problem is whether this theorem can be proved without relying on ax-12 1393 (see note in a12study 1825). Theorem ax11 1655 shows the reverse derivation of ax-11 1389 from ax-11o 1654. Normally, ax11o 1653 should be used rather than ax-11o 1654, except by theorems specifically studying the latter's properties. |
Axiom | ax-11o 1654 |
Axiom ax-11o 1654 ("o" for "old") was the
original version of ax-11 1389,
before it was discovered (in Jan. 2007) that the shorter ax-11 1389 could
replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of
the preprint). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of
[Monk2] p. 105, from which it can be proved
by cases. To understand this
theorem more easily, think of " ..." as informally
meaning "if
and are distinct
variables then..." The
antecedent becomes false if the same variable is substituted for and
, ensuring the
theorem is sound whenever this is the case. In some
later theorems, we call an antecedent of the form a
"distinctor."
This axiom is redundant, as shown by theorem ax11o 1653. Normally, ax11o 1653 should be used rather than ax-11o 1654, except by theorems specifically studying the latter's properties. |
Theorem | ax11 1655 |
Rederivation of axiom ax-11 1389 from the orginal version, ax-11o 1654. See
theorem ax11o 1653 for the derivation of ax-11o 1654 from ax-11 1389.
This theorem should not be referenced in any proof. Instead, use ax-11 1389 above so that uses of ax-11 1389 can be more easily identified. |
Theorem | ax11b 1656 | A bidirectional version of ax-11o 1654. |
Theorem | equs5 1657 | Lemma used in proofs of substitution properties. |
Theorem | sb3 1658 | One direction of a simplified definition of substitution when variables are distinct. |
Theorem | sb4 1659 | One direction of a simplified definition of substitution when variables are distinct. |
Theorem | sb4b 1660 | Simplified definition of substitution when variables are distinct. |
Theorem | dfsb2 1661 | An alternate definition of proper substitution that, like df-sb 1606, mixes free and bound variables to avoid distinct variable requirements. |
Theorem | dfsb3 1662 | An alternate definition of proper substitution df-sb 1606 that uses only primitive connectives (no defined terms) on the right-hand side. |
Theorem | hbsb2 1663 | Bound-variable hypothesis builder for substitution. |
Theorem | sbequi 1664 | An equality theorem for substitution. |
Theorem | sbequ 1665 | An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). |
Theorem | drsb2 1666 | Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). |
Theorem | sbn 1667 | Negation inside and outside of substitution are equivalent. |
Theorem | sbi1 1668 | Removal of implication from substitution. |
Theorem | sbi2 1669 | Introduction of implication into substitution. |
Theorem | sbim 1670 | Implication inside and outside of substitution are equivalent. |
Theorem | sbor 1671 | Logical OR inside and outside of substitution are equivalent. |
Theorem | sbrim 1672 | Substitution with a variable not free in antecedent affects only the consequent. |
Theorem | sblim 1673 | Substitution with a variable not free in consequent affects only the antecedent. |
Theorem | sban 1674 | Conjunction inside and outside of a substitution are equivalent. |
Theorem | sb3an 1675 | Conjunction inside and outside of a substitution are equivalent. |
Theorem | sbbi 1676 | Equivalence inside and outside of a substitution are equivalent. |
Theorem | sblbis 1677 | Introduce left biconditional inside of a substitution. |
Theorem | sbrbis 1678 | Introduce right biconditional inside of a substitution. |
Theorem | sbrbif 1679 | Introduce right biconditional inside of a substitution. |
Theorem | a4sbe 1680 | A specialization theorem. |
Theorem | a4sbim 1681 | Specialization of implication. (The proof was shortened by Andrew Salmon, 25-May-2011.) |
Theorem | a4sbbi 1682 | Specialization of biconditional. |
Theorem | sbbid 1683 | Deduction substituting both sides of a biconditional. |
Theorem | sbequ8 1684 | Elimination of equality from antecedent after substitution. |
Theorem | sbf3t 1685 | Substitution has no effect on a non-free variable. |
Theorem | hbsb4 1686 | A variable not free remains so after substitution with a distinct variable. |
Theorem | hbsb4t 1687 | A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1686). (The proof was shortened by Andrew Salmon, 25-May-2011.) |
Theorem | dvelimf 1688 | Version of dvelim 1799 without any variable restrictions. |
Theorem | dvelimdf 1689 | Deduction form of dvelimf 1688. This version may be useful if we want to avoid ax-17 1402 and use ax-16 1644 instead. |
Theorem | sbco 1690 | A composition law for substitution. |
Theorem | sbid2 1691 | An identity law for substitution. |
Theorem | sbidm 1692 | An idempotent law for substitution. (The proof was shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sbco2 1693 | A composition law for substitution. |
Theorem | sbco2d 1694 | A composition law for substitution. |
Theorem | sbco3 1695 | A composition law for substitution. |
Theorem | sbcom 1696 | A commutativity law for substitution. |
Theorem | sb5rf 1697 | Reversed substitution. (The proof was shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sb6rf 1698 | Reversed substitution. (The proof was shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sb8 1699 | Substitution of variable in universal quantifier. (The proof was shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sb8e 1700 | Substitution of variable in existential quantifier. |
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