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Type | Label | Description |
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Statement | ||
Theorem | xornbidc 1401 | Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.) |
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Theorem | xordc 1402 | Two ways to express "exclusive or" between decidable propositions. Theorem *5.22 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.) |
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Theorem | xordc1 1403 | Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.) |
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Theorem | nbbndc 1404 | Move negation outside of biconditional, for decidable propositions. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.) |
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Theorem | biassdc 1405 |
Associative law for the biconditional, for decidable propositions.
The classical version (without the decidability conditions) is an axiom of system DS in Vladimir Lifschitz, "On calculational proofs", Annals of Pure and Applied Logic, 113:207-224, 2002, http://www.cs.utexas.edu/users/ai-lab/pub-view.php?PubID=26805, and, interestingly, was not included in Principia Mathematica but was apparently first noted by Jan Lukasiewicz circa 1923. (Contributed by Jim Kingdon, 4-May-2018.) |
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Theorem | bilukdc 1406 | Lukasiewicz's shortest axiom for equivalential calculus (but modified to require decidable propositions). Storrs McCall, ed., Polish Logic 1920-1939 (Oxford, 1967), p. 96. (Contributed by Jim Kingdon, 5-May-2018.) |
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Theorem | dfbi3dc 1407 | An alternate definition of the biconditional for decidable propositions. Theorem *5.23 of [WhiteheadRussell] p. 124, but with decidability conditions. (Contributed by Jim Kingdon, 5-May-2018.) |
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Theorem | pm5.24dc 1408 | Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.) |
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Theorem | xordidc 1409 | Conjunction distributes over exclusive-or, for decidable propositions. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic. (Contributed by Jim Kingdon, 14-Jul-2018.) |
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Theorem | anxordi 1410 | Conjunction distributes over exclusive-or. (Contributed by Mario Carneiro and Jim Kingdon, 7-Oct-2018.) |
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For classical logic, truth tables can be used to define propositional
logic operations, by showing the results of those operations for all
possible combinations of true (
Although the intuitionistic logic connectives are not as simply defined,
Here we show that our definitions and axioms produce equivalent results for
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Theorem | truantru 1411 |
A ![]() |
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Theorem | truanfal 1412 |
A ![]() |
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Theorem | falantru 1413 |
A ![]() |
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Theorem | falanfal 1414 |
A ![]() |
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Theorem | truortru 1415 |
A ![]() |
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Theorem | truorfal 1416 |
A ![]() |
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Theorem | falortru 1417 |
A ![]() |
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Theorem | falorfal 1418 |
A ![]() |
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Theorem | truimtru 1419 |
A ![]() |
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Theorem | truimfal 1420 |
A ![]() |
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Theorem | falimtru 1421 |
A ![]() |
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Theorem | falimfal 1422 |
A ![]() |
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Theorem | nottru 1423 |
A ![]() |
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Theorem | notfal 1424 |
A ![]() |
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Theorem | trubitru 1425 |
A ![]() |
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Theorem | trubifal 1426 |
A ![]() |
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Theorem | falbitru 1427 |
A ![]() |
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Theorem | falbifal 1428 |
A ![]() |
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Theorem | truxortru 1429 |
A ![]() |
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Theorem | truxorfal 1430 |
A ![]() |
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Theorem | falxortru 1431 |
A ![]() |
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Theorem | falxorfal 1432 |
A ![]() |
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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 663, modus ponendo tollens I mptnan 1433, modus ponendo tollens II mptxor 1434, and modus tollendo ponens (exclusive-or version) mtpxor 1436. 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 1436 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 1435. This set of indemonstrables is not the entire system of Stoic logic. | ||
Theorem | mptnan 1433 | 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 1434) 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.) |
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Theorem | mptxor 1434 |
Modus ponendo tollens 2, one of the "indemonstrables" in Stoic logic.
Note that this uses exclusive-or ![]() |
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Theorem | mtpor 1435 |
Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism.
This is similar to mtpxor 1436, 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 ![]() ![]() ![]() ![]() |
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Theorem | mtpxor 1436 |
Modus tollendo ponens (original exclusive-or version), aka disjunctive
syllogism, similar to mtpor 1435, one of the five "indemonstrables"
in
Stoic logic. The rule says, "if ![]() ![]() ![]() ![]() |
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Theorem | stoic1a 1437 |
Stoic logic Thema 1 (part a).
The first thema of the four Stoic logic themata, in its basic form, was: "When from two (assertibles) a third follows, then from either of them together with the contradictory of the conclusion the contradictory of the other follows." (Apuleius Int. 209.9-14), see [Bobzien] p. 117 and https://plato.stanford.edu/entries/logic-ancient/ We will represent thema 1 as two very similar rules stoic1a 1437 and stoic1b 1438 to represent each side. (Contributed by David A. Wheeler, 16-Feb-2019.) (Proof shortened by Wolf Lammen, 21-May-2020.) |
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Theorem | stoic1b 1438 | Stoic logic Thema 1 (part b). The other part of thema 1 of Stoic logic; see stoic1a 1437. (Contributed by David A. Wheeler, 16-Feb-2019.) |
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Theorem | stoic2a 1439 |
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 This version a is without the phrase "or both"; see stoic2b 1440 for the version with the phrase "or both". We already have this rule as syldan 282, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.) |
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Theorem | stoic2b 1440 |
Stoic logic Thema 2 version b. See stoic2a 1439.
Version b is with the phrase "or both". We already have this rule as mpd3an3 1348, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.) |
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Theorem | stoic3 1441 |
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.) |
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Theorem | stoic4a 1442 |
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 |
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Theorem | stoic4b 1443 |
Stoic logic Thema 4 version b.
This is version b, which is with the phrase "or both". See stoic4a 1442 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.) |
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Theorem | syl6an 1444 | A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.) |
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Theorem | syl10 1445 | A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.) |
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Theorem | exbir 1446 | Exportation implication also converting head from biconditional to conditional. (Contributed by Alan Sare, 31-Dec-2011.) |
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Theorem | 3impexp 1447 | impexp 263 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.) |
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Theorem | 3impexpbicom 1448 | 3impexp 1447 with biconditional consequent of antecedent that is commuted in consequent. (Contributed by Alan Sare, 31-Dec-2011.) |
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Theorem | 3impexpbicomi 1449 | Deduction form of 3impexpbicom 1448. (Contributed by Alan Sare, 31-Dec-2011.) |
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Theorem | ancomsimp 1450 | Closed form of ancoms 268. (Contributed by Alan Sare, 31-Dec-2011.) |
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Theorem | expcomd 1451 | Deduction form of expcom 116. (Contributed by Alan Sare, 22-Jul-2012.) |
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Theorem | expdcom 1452 | Commuted form of expd 258. (Contributed by Alan Sare, 18-Mar-2012.) |
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Theorem | simplbi2comg 1453 | Implication form of simplbi2com 1454. (Contributed by Alan Sare, 22-Jul-2012.) |
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Theorem | simplbi2com 1454 | A deduction eliminating a conjunct, similar to simplbi2 385. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.) |
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Theorem | syl6ci 1455 | A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.) |
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Theorem | mpisyl 1456 | A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.) |
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The universal quantifier was introduced above in wal 1361 for use by df-tru 1366. See the comments in that section. In this section, we continue with the first "real" use of it. | ||
Axiom | ax-5 1457 | Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) |
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Axiom | ax-7 1458 | 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.) |
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Axiom | ax-gen 1459 |
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 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | gen2 1460 | Generalization applied twice. (Contributed by NM, 30-Apr-1998.) |
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Theorem | mpg 1461 | Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.) |
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Theorem | mpgbi 1462 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
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Theorem | mpgbir 1463 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
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Theorem | a7s 1464 | Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.) |
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Theorem | alimi 1465 | Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 2alimi 1466 | Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.) |
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Theorem | alim 1467 | Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 30-Mar-2008.) |
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Theorem | al2imi 1468 | Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.) |
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Theorem | alanimi 1469 | Variant of al2imi 1468 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.) |
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Syntax | wnf 1470 | Extend wff definition to include the not-free predicate. |
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Definition | df-nf 1471 |
Define the not-free predicate for wffs. This is read "![]() ![]() ![]() ![]() ![]() ![]() Nonfreeness is a commonly used condition, 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 notion of nonfreeness 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 considers syntactic freedom). For example,
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Theorem | nfi 1472 |
Deduce that ![]() ![]() |
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Theorem | hbth 1473 |
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
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Theorem | nfth 1474 | No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | nfnth 1475 | No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) |
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Theorem | nftru 1476 | The true constant has no free variables. (This can also be proven in one step with nfv 1538, but this proof does not use ax-17 1536.) (Contributed by Mario Carneiro, 6-Oct-2016.) |
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Theorem | alimdh 1477 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 4-Jan-2002.) |
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Theorem | albi 1478 | Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | alrimih 1479 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
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Theorem | albii 1480 | Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.) |
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Theorem | 2albii 1481 | Inference adding 2 universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.) |
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Theorem | hbxfrbi 1482 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
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Theorem | nfbii 1483 | Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | nfxfr 1484 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | nfxfrd 1485 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | alcoms 1486 | Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.) |
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Theorem | hbal 1487 |
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Theorem | alcom 1488 | Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
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Theorem | alrimdh 1489 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
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Theorem | albidh 1490 | Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.26 1491 | 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.) |
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Theorem | 19.26-2 1492 | Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.) |
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Theorem | 19.26-3an 1493 | Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.) |
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Theorem | 19.33 1494 | Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | alrot3 1495 | Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | alrot4 1496 | Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 28-Jun-2014.) |
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Theorem | albiim 1497 | Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.) |
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Theorem | 2albiim 1498 | Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.) |
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Theorem | hband 1499 | Deduction form of bound-variable hypothesis builder hban 1557. (Contributed by NM, 2-Jan-2002.) |
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Theorem | hb3and 1500 | Deduction form of bound-variable hypothesis builder hb3an 1560. (Contributed by NM, 17-Feb-2013.) |
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