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Statement | ||
Theorem | falxorfal 1401 | A identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
The Greek Stoics developed a system of logic. The Stoic Chrysippus, in particular, was often considered one of the greatest logicians of antiquity. Stoic logic is different from Aristotle's system, since it focuses on propositional logic, though later thinkers did combine the systems of the Stoics with Aristotle. Jan Lukasiewicz reports, "For anybody familiar with mathematical logic it is self-evident that the Stoic dialectic is the ancient form of modern propositional logic" ( On the history of the logic of proposition by Jan Lukasiewicz (1934), translated in: Selected Works - Edited by Ludwik Borkowski - Amsterdam, North-Holland, 1970 pp. 197-217, referenced in "History of Logic" https://www.historyoflogic.com/logic-stoics.htm). For more about Aristotle's system, see barbara and related theorems. A key part of the Stoic logic system is a set of five "indemonstrables" assigned to Chrysippus of Soli by Diogenes Laertius, though in general it is difficult to assign specific ideas to specific thinkers. The indemonstrables are described in, for example, [Lopez-Astorga] p. 11 , [Sanford] p. 39, and [Hitchcock] p. 5. These indemonstrables are modus ponendo ponens (modus ponens) ax-mp 5, modus tollendo tollens (modus tollens) mto 652, modus ponendo tollens I mptnan 1402, modus ponendo tollens II mptxor 1403, and modus tollendo ponens (exclusive-or version) mtpxor 1405. The first is an axiom, the second is already proved; in this section we prove the other three. Since we assume or prove all of indemonstrables, the system of logic we use here is as at least as strong as the set of Stoic indemonstrables. Note that modus tollendo ponens mtpxor 1405 originally used exclusive-or, but over time the name modus tollendo ponens has increasingly referred to an inclusive-or variation, which is proved in mtpor 1404. This set of indemonstrables is not the entire system of Stoic logic. | ||
Theorem | mptnan 1402 | Modus ponendo tollens 1, one of the "indemonstrables" in Stoic logic. See rule 1 on [Lopez-Astorga] p. 12 , rule 1 on [Sanford] p. 40, and rule A3 in [Hitchcock] p. 5. Sanford describes this rule second (after mptxor 1403) as a "safer, and these days much more common" version of modus ponendo tollens because it avoids confusion between inclusive-or and exclusive-or. (Contributed by David A. Wheeler, 3-Jul-2016.) |
Theorem | mptxor 1403 | Modus ponendo tollens 2, one of the "indemonstrables" in Stoic logic. Note that this uses exclusive-or . See rule 2 on [Lopez-Astorga] p. 12 , rule 4 on [Sanford] p. 39 and rule A4 in [Hitchcock] p. 5 . (Contributed by David A. Wheeler, 2-Mar-2018.) |
Theorem | mtpor 1404 | Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1405, one of the five original "indemonstrables" in Stoic logic. However, in Stoic logic this rule used exclusive-or, while the name modus tollendo ponens often refers to a variant of the rule that uses inclusive-or instead. The rule says, "if is not true, and or (or both) are true, then must be true." An alternate phrasing is, "Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth." -- Sherlock Holmes (Sir Arthur Conan Doyle, 1890: The Sign of the Four, ch. 6). (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.) |
Theorem | mtpxor 1405 | Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1404, one of the five "indemonstrables" in Stoic logic. The rule says, "if is not true, and either or (exclusively) are true, then must be true." Today the name "modus tollendo ponens" often refers to a variant, the inclusive-or version as defined in mtpor 1404. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1403, that is, it is exclusive-or df-xor 1355), rule 3 of [Sanford] p. 39 (where it is not as clearly stated which kind of "or" is used but it appears to be in the same sense as mptxor 1403), and rule A5 in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by David A. Wheeler, 4-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.) (Proof shortened by BJ, 19-Apr-2019.) |
Theorem | stoic2a 1406 |
Stoic logic Thema 2 version a.
Statement T2 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic thema 2 as follows: "When from two assertibles a third follows, and from the third and one (or both) of the two another follows, then this other follows from the first two." Bobzien uses constructs such as , ; in Metamath we will represent that construct as . This version a is without the phrase "or both"; see stoic2b 1407 for the version with the phrase "or both". We already have this rule as syldan 280, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic2b 1407 |
Stoic logic Thema 2 version b. See stoic2a 1406.
Version b is with the phrase "or both". We already have this rule as mpd3an3 1317, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic3 1408 |
Stoic logic Thema 3.
Statement T3 of [Bobzien] p. 116-117 discusses Stoic logic thema 3. "When from two (assemblies) a third follows, and from the one that follows (i.e., the third) together with another, external external assumption, another follows, then other follows from the first two and the externally co-assumed one. (Simp. Cael. 237.2-4)" (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic4a 1409 |
Stoic logic Thema 4 version a.
Statement T4 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic thema 4: "When from two assertibles a third follows, and from the third and one (or both) of the two and one (or more) external assertible(s) another follows, then this other follows from the first two and the external(s)." We use to represent the "external" assertibles. This is version a, which is without the phrase "or both"; see stoic4b 1410 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic4b 1410 |
Stoic logic Thema 4 version b.
This is version b, which is with the phrase "or both". See stoic4a 1409 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | syl6an 1411 | A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.) |
Theorem | syl10 1412 | A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.) |
Theorem | exbir 1413 | Exportation implication also converting head from biconditional to conditional. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | 3impexp 1414 | impexp 261 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | 3impexpbicom 1415 | 3impexp 1414 with biconditional consequent of antecedent that is commuted in consequent. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | 3impexpbicomi 1416 | Deduction form of 3impexpbicom 1415. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | ancomsimp 1417 | Closed form of ancoms 266. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | expcomd 1418 | Deduction form of expcom 115. (Contributed by Alan Sare, 22-Jul-2012.) |
Theorem | expdcom 1419 | Commuted form of expd 256. (Contributed by Alan Sare, 18-Mar-2012.) |
Theorem | simplbi2comg 1420 | Implication form of simplbi2com 1421. (Contributed by Alan Sare, 22-Jul-2012.) |
Theorem | simplbi2com 1421 | A deduction eliminating a conjunct, similar to simplbi2 383. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.) |
Theorem | syl6ci 1422 | A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.) |
Theorem | mpisyl 1423 | A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.) |
The universal quantifier was introduced above in wal 1330 for use by df-tru 1335. See the comments in that section. In this section, we continue with the first "real" use of it. | ||
Axiom | ax-5 1424 | Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) |
Axiom | ax-7 1425 | Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. One of the predicate logic axioms which do not involve equality. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. (Contributed by NM, 5-Aug-1993.) |
Axiom | ax-gen 1426 | Rule of Generalization. The postulated inference rule of predicate calculus. See e.g. Rule 2 of [Hamilton] p. 74. This rule says that if something is unconditionally true, then it is true for all values of a variable. For example, if we have proved , we can conclude or even . Theorem spi 1513 shows we can go the other way also: in other words we can add or remove universal quantifiers from the beginning of any theorem as required. (Contributed by NM, 5-Aug-1993.) |
Theorem | gen2 1427 | Generalization applied twice. (Contributed by NM, 30-Apr-1998.) |
Theorem | mpg 1428 | Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.) |
Theorem | mpgbi 1429 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
Theorem | mpgbir 1430 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
Theorem | a7s 1431 | Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.) |
Theorem | alimi 1432 | Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Aug-1993.) |
Theorem | 2alimi 1433 | Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.) |
Theorem | alim 1434 | Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 30-Mar-2008.) |
Theorem | al2imi 1435 | Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.) |
Theorem | alanimi 1436 | Variant of al2imi 1435 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.) |
Syntax | wnf 1437 | Extend wff definition to include the not-free predicate. |
Definition | df-nf 1438 |
Define the not-free predicate for wffs. This is read " is not free
in ".
Not-free means that the value of cannot affect the
value of ,
e.g., any occurrence of in
is effectively
bound by a "for all" or something that expands to one (such as
"there
exists"). In particular, substitution for a variable not free in a
wff
does not affect its value (sbf 1746). An example of where this is used is
stdpc5 1560. See nf2 1644 for an alternate definition which
does not involve
nested quantifiers on the same variable.
Not-free is a commonly used constraint, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the not-free notion within the logic itself rather than as a metalogical side condition. To be precise, our definition really means "effectively not free," because it is slightly less restrictive than the usual textbook definition for not-free (which only considers syntactic freedom). For example, is effectively not free in the bare expression , even though would be considered free in the usual textbook definition, because the value of in the expression cannot affect the truth of the expression (and thus substitution will not change the result). (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfi 1439 | Deduce that is not free in from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbth 1440 |
No variable is (effectively) free in a theorem.
This and later "hypothesis-building" lemmas, with labels starting "hb...", allow us to construct proofs of formulas of the form from smaller formulas of this form. These are useful for constructing hypotheses that state " is (effectively) not free in ." (Contributed by NM, 5-Aug-1993.) |
Theorem | nfth 1441 | No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfnth 1442 | No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) |
Theorem | nftru 1443 | The true constant has no free variables. (This can also be proven in one step with nfv 1505, but this proof does not use ax-17 1503.) (Contributed by Mario Carneiro, 6-Oct-2016.) |
Theorem | alimdh 1444 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 4-Jan-2002.) |
Theorem | albi 1445 | Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | alrimih 1446 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | albii 1447 | Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.) |
Theorem | 2albii 1448 | Inference adding 2 universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.) |
Theorem | hbxfrbi 1449 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Theorem | nfbii 1450 | Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfxfr 1451 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfxfrd 1452 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | alcoms 1453 | Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.) |
Theorem | hbal 1454 | If is not free in , it is not free in . (Contributed by NM, 5-Aug-1993.) |
Theorem | alcom 1455 | Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
Theorem | alrimdh 1456 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | albidh 1457 | Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.26 1458 | Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) |
Theorem | 19.26-2 1459 | Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.) |
Theorem | 19.26-3an 1460 | Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.) |
Theorem | 19.33 1461 | Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | alrot3 1462 | Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |
Theorem | alrot4 1463 | Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 28-Jun-2014.) |
Theorem | albiim 1464 | Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.) |
Theorem | 2albiim 1465 | Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.) |
Theorem | hband 1466 | Deduction form of bound-variable hypothesis builder hban 1523. (Contributed by NM, 2-Jan-2002.) |
Theorem | hb3and 1467 | Deduction form of bound-variable hypothesis builder hb3an 1526. (Contributed by NM, 17-Feb-2013.) |
Theorem | hbald 1468 | Deduction form of bound-variable hypothesis builder hbal 1454. (Contributed by NM, 2-Jan-2002.) |
Syntax | wex 1469 | Extend wff definition to include the existential quantifier ("there exists"). |
Axiom | ax-ie1 1470 | is bound in . One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Axiom | ax-ie2 1471 | Define existential quantification. means "there exists at least one set such that is true." One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Theorem | hbe1 1472 | is not free in . (Contributed by NM, 5-Aug-1993.) |
Theorem | nfe1 1473 | is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | 19.23ht 1474 | Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 1-Feb-2015.) |
Theorem | 19.23h 1475 | Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.) |
Theorem | alnex 1476 | Theorem 19.7 of [Margaris] p. 89. To read this intuitionistically, think of it as "if can be refuted for all , then it is not possible to find an for which holds" (and likewise for the converse). Comparing this with dfexdc 1478 illustrates that statements which look similar (to someone used to classical logic) can be different intuitionistically due to different placement of negations. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 1-Feb-2015.) (Revised by Mario Carneiro, 12-May-2015.) |
Theorem | nex 1477 | Generalization rule for negated wff. (Contributed by NM, 18-May-1994.) |
Theorem | dfexdc 1478 | Defining given decidability. It is common in classical logic to define as but in intuitionistic logic without a decidability condition, that is only an implication not an equivalence, as seen at exalim 1479. (Contributed by Jim Kingdon, 15-Mar-2018.) |
DECID | ||
Theorem | exalim 1479 | One direction of a classical definition of existential quantification. One direction of Definition of [Margaris] p. 49. For a decidable proposition, this is an equivalence, as seen as dfexdc 1478. (Contributed by Jim Kingdon, 29-Jul-2018.) |
The equality predicate was introduced above in wceq 1332 for use by df-tru 1335. See the comments in that section. In this section, we continue with the first "real" use of it. | ||
Theorem | weq 1480 |
Extend wff definition to include atomic formulas using the equality
predicate.
(Instead of introducing weq 1480 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 1332. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1480 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1332. 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.) |
Axiom | ax-8 1481 |
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 1683). Axiom scheme C8'
in [Megill] p. 448 (p. 16 of the preprint).
Also appears as Axiom C7 of
[Monk2] p. 105.
Axioms ax-8 1481 through ax-16 1782 are the axioms having to do with equality, substitution, and logical properties of our binary predicate (which later in set theory will mean "is a member of"). Note that all axioms except ax-16 1782 and ax-17 1503 are still valid even when , , and are replaced with the same variable because they do not have any distinct variable (Metamath's $d) restrictions. Distinct variable restrictions are required for ax-16 1782 and ax-17 1503 only. (Contributed by NM, 5-Aug-1993.) |
Axiom | ax-10 1482 |
Axiom of Quantifier Substitution. One of the equality and substitution
axioms of predicate calculus with equality. Appears as Lemma L12 in
[Megill] p. 445 (p. 12 of the preprint).
The original version of this axiom was ax-10o 1690 ("o" for "old") and was replaced with this shorter ax-10 1482 in May 2008. The old axiom is proved from this one as theorem ax10o 1689. Conversely, this axiom is proved from ax-10o 1690 as theorem ax10 1691. (Contributed by NM, 5-Aug-1993.) |
Axiom | ax-11 1483 |
Axiom of Variable Substitution. One of the 5 equality axioms of predicate
calculus. The final consequent
is a way of
expressing "
substituted for in wff
" (cf. sb6 1854).
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.
Variants of this axiom which are equivalent in classical logic but which have not been shown to be equivalent for intuitionistic logic are ax11v 1795, ax11v2 1788 and ax-11o 1791. (Contributed by NM, 5-Aug-1993.) |
Axiom | ax-i12 1484 |
Axiom of Quantifier Introduction. One of the equality and substitution
axioms of predicate calculus with equality. Informally, it says that
whenever is
distinct from and , and is
true,
then quantified with is also true. In other words,
is irrelevant to the truth of . Axiom scheme C9' 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.
This axiom has been modified from the original ax12 1489 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias ax12or 1485 instead, for labeling consistency. (New usage is discouraged.) |
Theorem | ax12or 1485 | Alias for ax-i12 1484, to be used in place of it for labeling consistency. (Contributed by NM, 3-Feb-2015.) |
Axiom | ax-bndl 1486 |
Axiom of bundling. The general idea of this axiom is that two variables
are either distinct or non-distinct. That idea could be expressed as
. However, we instead choose an axiom
which has many of the same consequences, but which is different with
respect to a universe which contains only one object. holds
if and are the same variable,
likewise for and ,
and
holds if is distinct from
the others (and the universe has at least two objects).
As with other statements of the form "x is decidable (either true or false)", this does not entail the full Law of the Excluded Middle (which is the proposition that all statements are decidable), but instead merely the assertion that particular kinds of statements are decidable (or in this case, an assertion similar to decidability). This axiom implies ax-i12 1484 as can be seen at axi12 1491. Whether ax-bndl 1486 can be proved from the remaining axioms including ax-i12 1484 is not known. The reason we call this "bundling" is that a statement without a distinct variable constraint "bundles" together two statements, one in which the two variables are the same and one in which they are different. (Contributed by Mario Carneiro and Jim Kingdon, 14-Mar-2018.) |
Axiom | ax-4 1487 |
Axiom of Specialization. A quantified wff implies the wff without a
quantifier (i.e. an instance, or special case, of the generalized wff).
In other words if something is true for all , it is true for any
specific (that
would typically occur as a free variable in the wff
substituted for ). (A free variable is one that does not occur in
the scope of a quantifier: and are both
free in ,
but only is free
in .) Axiom
scheme C5' in [Megill]
p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski]
p. 67 (under his system S2, defined in the last paragraph on p. 77).
Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1426. Conditional forms of the converse are given by ax12 1489, ax-16 1782, and ax-17 1503. Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 1744. (Contributed by NM, 5-Aug-1993.) |
Theorem | sp 1488 | Specialization. Another name for ax-4 1487. (Contributed by NM, 21-May-2008.) |
Theorem | ax12 1489 | Rederive the original version of the axiom from ax-i12 1484. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Theorem | hbequid 1490 |
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.
The proof uses only ax-8 1481 and ax-i12 1484 on top of (the FOL analogue of) modal logic KT. This shows that this can be proved without ax-i9 1507, even though the theorem equid 1675 cannot. A shorter proof using ax-i9 1507 is obtainable from equid 1675 and hbth 1440. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) |
Theorem | axi12 1491 | Proof that ax-i12 1484 follows from ax-bndl 1486. So that we can track which theorems rely on ax-bndl 1486, proofs should reference ax12or 1485 rather than this theorem. (Contributed by Jim Kingdon, 17-Aug-2018.) (New usage is discouraged). (Proof modification is discouraged.) |
Theorem | alequcom 1492 | Commutation law for identical variable specifiers. The antecedent and consequent are true when and are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.) |
Theorem | alequcoms 1493 | A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.) |
Theorem | nalequcoms 1494 | A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 2-Feb-2015.) |
Theorem | nfr 1495 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) |
Theorem | nfri 1496 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfrd 1497 | Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | alimd 1498 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | alrimi 1499 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | nfd 1500 | Deduce that is not free in in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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