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

Theorem19.12 1601 Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! (Contributed by NM, 5-Aug-1993.)

Theorem19.19 1602 Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Theorem19.21-2 1603 Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.)

Theoremnf2 1604 An alternate definition of df-nf 1396, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnf3 1605 An alternate definition of df-nf 1396. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnf4dc 1606 Variable is effectively not free in iff is always true or always false, given a decidability condition. The reverse direction, nf4r 1607, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.)
DECID

Theoremnf4r 1607 If is always true or always false, then variable is effectively not free in . The converse holds given a decidability condition, as seen at nf4dc 1606. (Contributed by Jim Kingdon, 21-Jul-2018.)

Theorem19.36i 1608 Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)

Theorem19.36-1 1609 Closed form of 19.36i 1608. One direction of Theorem 19.36 of [Margaris] p. 90. The converse holds in classical logic, but does not hold (for all propositions) in intuitionistic logic. (Contributed by Jim Kingdon, 20-Jun-2018.)

Theorem19.37-1 1610 One direction of Theorem 19.37 of [Margaris] p. 90. The converse holds in classical logic but not, in general, here. (Contributed by Jim Kingdon, 21-Jun-2018.)

Theorem19.37aiv 1611* Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.38 1612 Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.23t 1613 Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

Theorem19.23 1614 Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Theorem19.32dc 1615 Theorem 19.32 of [Margaris] p. 90, where is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
DECID

Theorem19.32r 1616 One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if is decidable, as seen at 19.32dc 1615. (Contributed by Jim Kingdon, 28-Jul-2018.)

Theorem19.31r 1617 One direction of Theorem 19.31 of [Margaris] p. 90. The converse holds in classical logic, but not intuitionistic logic. (Contributed by Jim Kingdon, 28-Jul-2018.)

Theorem19.44 1618 Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Theorem19.45 1619 Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Theorem19.34 1620 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.41h 1621 Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1622 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)

Theorem19.41 1622 Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)

Theorem19.42h 1623 Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1624 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.)

Theorem19.42 1624 Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)

Theoremexcom13 1625 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)

Theoremexrot3 1626 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)

Theoremexrot4 1627 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)

Theoremnexr 1628 Inference from 19.8a 1528. (Contributed by Jeff Hankins, 26-Jul-2009.)

Theoremexan 1629 Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremhbexd 1630 Deduction form of bound-variable hypothesis builder hbex 1573. (Contributed by NM, 2-Jan-2002.)

Theoremeeor 1631 Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)

1.3.8  Equality theorems without distinct variables

Theorema9e 1632 At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-5 1382 through ax-14 1451 and ax-17 1465, all axioms other than ax-9 1470 are believed to be theorems of free logic, although the system without ax-9 1470 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)

Theorema9ev 1633* At least one individual exists. Weaker version of a9e 1632. (Contributed by NM, 3-Aug-2017.)

Theoremax9o 1634 An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)

Theoremequid 1635 Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms.

This proof is similar to Tarski's and makes use of a dummy variable . It also works in intuitionistic logic, unlike some other possible ways of proving this theorem. (Contributed by NM, 1-Apr-2005.)

Theoremnfequid 1636 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.)

Theoremstdpc6 1637 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1701.) 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.)

Theoremequcomi 1638 Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.)

Theoremax6evr 1639* A commuted form of a9ev 1633. The naming reflects how axioms were numbered in the Metamath Proof Explorer as of 2020 (a numbering which we eventually plan to adopt here too, but until this happens everywhere only some theorems will have it). (Contributed by BJ, 7-Dec-2020.)

Theoremequcom 1640 Commutative law for equality. (Contributed by NM, 20-Aug-1993.)

Theoremequcomd 1641 Deduction form of equcom 1640, symmetry of equality. For the versions for classes, see eqcom 2091 and eqcomd 2094. (Contributed by BJ, 6-Oct-2019.)

Theoremequcoms 1642 An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.)

Theoremequtr 1643 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)

Theoremequtrr 1644 A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)

Theoremequtr2 1645 A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremequequ1 1646 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)

Theoremequequ2 1647 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)

Theoremelequ1 1648 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)

Theoremelequ2 1649 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)

Theoremax11i 1650 Inference that has ax-11 1443 (without ) as its conclusion and doesn't require ax-10 1442, ax-11 1443, or ax-12 1448 for its proof. 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.)

1.3.9  Axioms ax-10 and ax-11

Theoremax10o 1651 Show that ax-10o 1652 can be derived from ax-10 1442. An open problem is whether this theorem can be derived from ax-10 1442 and the others when ax-11 1443 is replaced with ax-11o 1752. See theorem ax10 1653 for the rederivation of ax-10 1442 from ax10o 1651.

Normally, ax10o 1651 should be used rather than ax-10o 1652, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)

Axiomax-10o 1652 Axiom ax-10o 1652 ("o" for "old") was the original version of ax-10 1442, before it was discovered (in May 2008) that the shorter ax-10 1442 could replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of the preprint).

This axiom is redundant, as shown by theorem ax10o 1651.

Normally, ax10o 1651 should be used rather than ax-10o 1652, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremax10 1653 Rederivation of ax-10 1442 from original version ax-10o 1652. See theorem ax10o 1651 for the derivation of ax-10o 1652 from ax-10 1442.

This theorem should not be referenced in any proof. Instead, use ax-10 1442 above so that uses of ax-10 1442 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

Theoremhbae 1654 All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)

Theoremnfae 1655 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbaes 1656 Rule that applies hbae 1654 to antecedent. (Contributed by NM, 5-Aug-1993.)

Theoremhbnae 1657 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.)

Theoremnfnae 1658 All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbnaes 1659 Rule that applies hbnae 1657 to antecedent. (Contributed by NM, 5-Aug-1993.)

Theoremnaecoms 1660 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)

Theoremequs4 1661 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)

Theoremequsalh 1662 A useful equivalence related to substitution. New proofs should use equsal 1663 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)

Theoremequsal 1663 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)

Theoremequsex 1664 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)

Theoremequsexd 1665 Deduction form of equsex 1664. (Contributed by Jim Kingdon, 29-Dec-2017.)

Theoremdral1 1666 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 24-Nov-1994.)

Theoremdral2 1667 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)

Theoremdrex2 1668 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)

Theoremdrnf1 1669 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)

Theoremdrnf2 1670 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)

Theoremspimth 1671 Closed theorem form of spim 1674. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.)

Theoremspimt 1672 Closed theorem form of spim 1674. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.)

Theoremspimh 1673 Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1674 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 8-May-2008.) (New usage is discouraged.)

Theoremspim 1674 Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1674 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)

Theoremspimeh 1675 Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by NM, 3-Feb-2015.) (New usage is discouraged.)

Theoremspimed 1676 Deduction version of spime 1677. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.)

Theoremspime 1677 Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.)

Theoremcbv3 1678 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)

Theoremcbv3h 1679 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.)

Theoremcbv1 1680 Rule used to change bound variables, using implicit substitution. Revised to format hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)

Theoremcbv1h 1681 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.)

Theoremcbv2h 1682 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Theoremcbv2 1683 Rule used to change bound variables, using implicit substitution. Revised to align format of hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)

Theoremcbvalh 1684 Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremcbval 1685 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremcbvexh 1686 Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.)

Theoremcbvex 1687 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Theoremchvar 1688 Implicit substitution of for into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremequvini 1689 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). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremequveli 1690 A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1689.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.)

Theoremnfald 1691 If is not free in , it is not free in . (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.)

Theoremnfexd 1692 If is not free in , it is not free in . (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 7-Feb-2018.)

1.3.10  Substitution (without distinct variables)

Syntaxwsb 1693 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 1694 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 1706.

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 1769, sbcom2 1912 and sbid2v 1921).

Note that our definition is valid even when and are replaced with the same variable, as sbid 1705 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 1916 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another alternate definition which uses a dummy variable is dfsb7a 1919.

When and are distinct, we can express proper substitution with the simpler expressions of sb5 1816 and sb6 1815.

In classical logic, another possible definition is but we do not have an intuitionistic proof that this is equivalent.

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

Theoremsbimi 1695 Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)

Theoremsbbii 1696 Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theoremsb1 1697 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsb2 1698 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ1 1699 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ2 1700 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

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