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Theorem List for Intuitionistic Logic Explorer - 1301-1400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmp3and 1301 A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑 → ((𝜓𝜒𝜃) → 𝜏))       (𝜑𝜏)

Theoremmp3an12i 1302 mp3an 1298 with antecedents in standard conjunction form and with one hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.)
𝜑    &   𝜓    &   (𝜒𝜃)    &   ((𝜑𝜓𝜃) → 𝜏)       (𝜒𝜏)

Theoremmp3an2i 1303 mp3an 1298 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
𝜑    &   (𝜓𝜒)    &   (𝜓𝜃)    &   ((𝜑𝜒𝜃) → 𝜏)       (𝜓𝜏)

Theoremmp3an3an 1304 mp3an 1298 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
𝜑    &   (𝜓𝜒)    &   (𝜃𝜏)    &   ((𝜑𝜒𝜏) → 𝜂)       ((𝜓𝜃) → 𝜂)

Theoremmp3an2ani 1305 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
𝜑    &   (𝜓𝜒)    &   ((𝜓𝜃) → 𝜏)    &   ((𝜑𝜒𝜏) → 𝜂)       ((𝜓𝜃) → 𝜂)

Theorembiimp3a 1306 Infer implication from a logical equivalence. Similar to biimpa 292. (Contributed by NM, 4-Sep-2005.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜓𝜒) → 𝜃)

Theorembiimp3ar 1307 Infer implication from a logical equivalence. Similar to biimpar 293. (Contributed by NM, 2-Jan-2009.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜓𝜃) → 𝜒)

Theorem3anandis 1308 Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.)
(((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜃)) → 𝜏)       ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)

Theorem3anandirs 1309 Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.) (Revised by NM, 18-Apr-2007.)
(((𝜑𝜃) ∧ (𝜓𝜃) ∧ (𝜒𝜃)) → 𝜏)       (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)

Theoremecased 1310 Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜓)

Theoremecase23d 1311 Variation of ecased 1310 with three disjuncts instead of two. (Contributed by NM, 22-Apr-1994.) (Revised by Jim Kingdon, 9-Dec-2017.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → ¬ 𝜃)    &   (𝜑 → (𝜓𝜒𝜃))       (𝜑𝜓)

1.2.12  True and false constants

1.2.12.1  Universal quantifier for use by df-tru

Even though it isn't ordinarily part of propositional calculus, the universal quantifier is introduced here so that the soundness of definition df-tru 1317 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in axiom ax-5 1406 in the predicate calculus section below. For those who want propositional calculus to be self-contained i.e. to use wff variables only, the alternate definition dftru2 1322 may be adopted and this subsection moved down to the start of the subsection with wex 1451 below. However, the use of dftru2 1322 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

Syntaxwal 1312 Extend wff definition to include the universal quantifier ('for all'). 𝑥𝜑 is read "𝜑 (phi) is true for all 𝑥." Typically, in its final application 𝜑 would be replaced with a wff containing a (free) occurrence of the variable 𝑥, for example 𝑥 = 𝑦. In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of 𝑥. When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same.
wff 𝑥𝜑

1.2.12.2  Equality predicate for use by df-tru

Even though it isn't ordinarily part of propositional calculus, the equality predicate = is introduced here so that the soundness of definition df-tru 1317 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in axiom ax-8 1465 in the predicate calculus section below. For those who want propositional calculus to be self-contained i.e. to use wff variables only, the alternate definition dftru2 1322 may be adopted and this subsection moved down to just above weq 1462 below. However, the use of dftru2 1322 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

Syntaxcv 1313 This syntax construction states that a variable 𝑥, which has been declared to be a setvar variable by \$f statement vx, is also a class expression. This can be justified informally as follows. We know that the class builder {𝑦𝑦𝑥} is a class by cab 2101. Since (when 𝑦 is distinct from 𝑥) we have 𝑥 = {𝑦𝑦𝑥} by cvjust 2110, we can argue that the syntax "class 𝑥 " can be viewed as an abbreviation for "class {𝑦𝑦𝑥}". 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 1313 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1313 is intrinsically no different from any other class-building syntax such as cab 2101, cun 3037, or c0 3331.

For a general discussion of the theory of classes and the role of cv 1313, see https://us.metamath.org/mpeuni/mmset.html#class 1313.

(The description above applies to set theory, not predicate calculus. The purpose of introducing class 𝑥 here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1462 of predicate calculus from the wceq 1314 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.)

class 𝑥

Syntaxwceq 1314 Extend wff definition to include class equality.

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

(The purpose of introducing wff 𝐴 = 𝐵 here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1462 of predicate calculus in terms of the wceq 1314 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 𝑥 = 𝑦 could be the = of either weq 1462 or wceq 1314, although mathematically it makes no difference. The class variables 𝐴 and 𝐵 are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 2108 for more information on the set theory usage of wceq 1314.)

wff 𝐴 = 𝐵

1.2.12.3  Define the true and false constants

Syntaxwtru 1315 is a wff.
wff

Theoremtrujust 1316 Soundness justification theorem for df-tru 1317. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.)
((∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥) ↔ (∀𝑦 𝑦 = 𝑦 → ∀𝑦 𝑦 = 𝑦))

Definitiondf-tru 1317 Definition of the truth value "true", or "verum", denoted by . This is a tautology, as proved by tru 1318. In this definition, an instance of id 19 is used as the definiens, although any tautology, such as an axiom, can be used in its place. This particular id 19 instance was chosen so this definition can be checked by the same algorithm that is used for predicate calculus. This definition should be referenced directly only by tru 1318, and other proofs should depend on tru 1318 (directly or indirectly) instead of this definition, since there are many alternate ways to define . (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by NM, 11-Jul-2019.) (New usage is discouraged.)
(⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))

Theoremtru 1318 The truth value is provable. (Contributed by Anthony Hart, 13-Oct-2010.)

Syntaxwfal 1319 is a wff.
wff

Definitiondf-fal 1320 Definition of the truth value "false", or "falsum", denoted by . See also df-tru 1317. (Contributed by Anthony Hart, 22-Oct-2010.)
(⊥ ↔ ¬ ⊤)

Theoremfal 1321 The truth value is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
¬ ⊥

Theoremdftru2 1322 An alternate definition of "true". (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) (New usage is discouraged.)
(⊤ ↔ (𝜑𝜑))

Theoremmptru 1323 Eliminate as an antecedent. A proposition implied by is true. (Contributed by Mario Carneiro, 13-Mar-2014.)
(⊤ → 𝜑)       𝜑

Theoremtbtru 1324 A proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.)
(𝜑 ↔ (𝜑 ↔ ⊤))

Theoremnbfal 1325 The negation of a proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.)
𝜑 ↔ (𝜑 ↔ ⊥))

Theorembitru 1326 A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
𝜑       (𝜑 ↔ ⊤)

Theorembifal 1327 A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
¬ 𝜑       (𝜑 ↔ ⊥)

Theoremfalim 1328 The truth value implies anything. Also called the principle of explosion, or "ex falso quodlibet". (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
(⊥ → 𝜑)

Theoremfalimd 1329 The truth value implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑 ∧ ⊥) → 𝜓)

Theorema1tru 1330 Anything implies . (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
(𝜑 → ⊤)

Theoremtruan 1331 True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
((⊤ ∧ 𝜑) ↔ 𝜑)

Theoremdfnot 1332 Given falsum, we can define the negation of a wff 𝜑 as the statement that a contradiction follows from assuming 𝜑. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
𝜑 ↔ (𝜑 → ⊥))

Theoreminegd 1333 Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑𝜓) → ⊥)       (𝜑 → ¬ 𝜓)

Theorempm2.21fal 1334 If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜓)       (𝜑 → ⊥)

Theorempclem6 1335 Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof rewritten by Jim Kingdon, 4-May-2018.)
((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓)

1.2.13  Logical 'xor'

Syntaxwxo 1336 Extend wff definition to include exclusive disjunction ('xor').
wff (𝜑𝜓)

Definitiondf-xor 1337 Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. Contrast with (wa 103), (wo 680), and (wi 4) . (Contributed by FL, 22-Nov-2010.) (Modified by Jim Kingdon, 1-Mar-2018.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))

Theoremxoranor 1338 One way of defining exclusive or. Equivalent to df-xor 1337. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)))

Theoremexcxor 1339 This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.) (Proof rewritten by Jim Kingdon, 5-May-2018.)
((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))

Theoremxoror 1340 XOR implies OR. (Contributed by BJ, 19-Apr-2019.)
((𝜑𝜓) → (𝜑𝜓))

Theoremxorbi2d 1341 Deduction joining an equivalence and a left operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))

Theoremxorbi1d 1342 Deduction joining an equivalence and a right operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))

Theoremxorbi12d 1343 Deduction joining two equivalences to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))

Theoremxorbi12i 1344 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))

Theoremxorbin 1345 A consequence of exclusive or. In classical logic the converse also holds. (Contributed by Jim Kingdon, 8-Mar-2018.)
((𝜑𝜓) → (𝜑 ↔ ¬ 𝜓))

Theorempm5.18im 1346 One direction of pm5.18dc 851, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.)
((𝜑𝜓) → ¬ (𝜑 ↔ ¬ 𝜓))

Theoremxornbi 1347 A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1352. (Contributed by Jim Kingdon, 10-Mar-2018.)
((𝜑𝜓) → ¬ (𝜑𝜓))

Theoremxor3dc 1348 Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))

Theoremxorcom 1349 is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.)
((𝜑𝜓) ↔ (𝜓𝜑))

Theorempm5.15dc 1350 A decidable proposition is equivalent to a decidable proposition or its negation. Based on theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ∨ (𝜑 ↔ ¬ 𝜓))))

Theoremxor2dc 1351 Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))))

Theoremxornbidc 1352 Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑𝜓))))

Theoremxordc 1353 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.)
(DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))))

Theoremxordc1 1354 Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.)
((𝜑𝜓) → DECID 𝜑)

Theoremnbbndc 1355 Move negation outside of biconditional, for decidable propositions. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ ¬ (𝜑𝜓))))

Theorembiassdc 1356 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.)

(DECID 𝜑 → (DECID 𝜓 → (DECID 𝜒 → (((𝜑𝜓) ↔ 𝜒) ↔ (𝜑 ↔ (𝜓𝜒))))))

Theorembilukdc 1357 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.)
(((DECID 𝜑DECID 𝜓) ∧ DECID 𝜒) → ((𝜑𝜓) ↔ ((𝜒𝜓) ↔ (𝜑𝜒))))

Theoremdfbi3dc 1358 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.)
(DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))))

Theorempm5.24dc 1359 Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.)
(DECID 𝜑 → (DECID 𝜓 → (¬ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))))

Theoremxordidc 1360 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.)
(DECID 𝜑 → (DECID 𝜓 → (DECID 𝜒 → ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒))))))

Theoremanxordi 1361 Conjunction distributes over exclusive-or. (Contributed by Mario Carneiro and Jim Kingdon, 7-Oct-2018.)
((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒)))

1.2.14  Truth tables: Operations on true and false constants

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 () and false ().

Although the intuitionistic logic connectives are not as simply defined, and do play similar roles as in classical logic and most theorems from classical logic continue to hold.

Here we show that our definitions and axioms produce equivalent results for and as we would get from truth tables for (conjunction aka logical 'and') wa 103, (disjunction aka logical inclusive 'or') wo 680, (implies) wi 4, ¬ (not) wn 3, (logical equivalence) df-bi 116, and (exclusive or) df-xor 1337.

Theoremtruantru 1362 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∧ ⊤) ↔ ⊤)

Theoremtruanfal 1363 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∧ ⊥) ↔ ⊥)

Theoremfalantru 1364 A identity. (Contributed by David A. Wheeler, 23-Feb-2018.)
((⊥ ∧ ⊤) ↔ ⊥)

Theoremfalanfal 1365 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∧ ⊥) ↔ ⊥)

Theoremtruortru 1366 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ∨ ⊤) ↔ ⊤)

Theoremtruorfal 1367 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∨ ⊥) ↔ ⊤)

Theoremfalortru 1368 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∨ ⊤) ↔ ⊤)

Theoremfalorfal 1369 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ∨ ⊥) ↔ ⊥)

Theoremtruimtru 1370 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ → ⊤) ↔ ⊤)

Theoremtruimfal 1371 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ → ⊥) ↔ ⊥)

Theoremfalimtru 1372 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ → ⊤) ↔ ⊤)

Theoremfalimfal 1373 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ → ⊥) ↔ ⊤)

Theoremnottru 1374 A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(¬ ⊤ ↔ ⊥)

Theoremnotfal 1375 A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(¬ ⊥ ↔ ⊤)

Theoremtrubitru 1376 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ↔ ⊤) ↔ ⊤)

Theoremtrubifal 1377 A identity. (Contributed by David A. Wheeler, 23-Feb-2018.)
((⊤ ↔ ⊥) ↔ ⊥)

Theoremfalbitru 1378 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ↔ ⊤) ↔ ⊥)

Theoremfalbifal 1379 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ↔ ⊥) ↔ ⊤)

Theoremtruxortru 1380 A identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
((⊤ ⊻ ⊤) ↔ ⊥)

Theoremtruxorfal 1381 A identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
((⊤ ⊻ ⊥) ↔ ⊤)

Theoremfalxortru 1382 A identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
((⊥ ⊻ ⊤) ↔ ⊤)

Theoremfalxorfal 1383 A identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
((⊥ ⊻ ⊥) ↔ ⊥)

1.2.15  Stoic logic indemonstrables (Chrysippus of Soli)

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 634, modus ponendo tollens I mptnan 1384, modus ponendo tollens II mptxor 1385, and modus tollendo ponens (exclusive-or version) mtpxor 1387. 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 1387 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 1386. This set of indemonstrables is not the entire system of Stoic logic.

Theoremmptnan 1384 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 1385) 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.)
𝜑    &    ¬ (𝜑𝜓)        ¬ 𝜓

Theoremmptxor 1385 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.)
𝜑    &   (𝜑𝜓)        ¬ 𝜓

Theoremmtpor 1386 Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1387, 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 𝜑 is not true, and 𝜑 or 𝜓 (or both) are true, then 𝜓 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.)
¬ 𝜑    &   (𝜑𝜓)       𝜓

Theoremmtpxor 1387 Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1386, one of the five "indemonstrables" in Stoic logic. The rule says, "if 𝜑 is not true, and either 𝜑 or 𝜓 (exclusively) are true, then 𝜓 must be true." Today the name "modus tollendo ponens" often refers to a variant, the inclusive-or version as defined in mtpor 1386. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1385, that is, it is exclusive-or df-xor 1337), 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 1385), 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.)
¬ 𝜑    &   (𝜑𝜓)       𝜓

Theoremstoic2a 1388 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 𝜑, 𝜓𝜒; in Metamath we will represent that construct as 𝜑𝜓𝜒.

This version a is without the phrase "or both"; see stoic2b 1389 for the version with the phrase "or both". We already have this rule as syldan 278, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)

((𝜑𝜓) → 𝜒)    &   ((𝜑𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremstoic2b 1389 Stoic logic Thema 2 version b. See stoic2a 1388.

Version b is with the phrase "or both". We already have this rule as mpd3an3 1299, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)

((𝜑𝜓) → 𝜒)    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremstoic3 1390 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.)

((𝜑𝜓) → 𝜒)    &   ((𝜒𝜃) → 𝜏)       ((𝜑𝜓𝜃) → 𝜏)

Theoremstoic4a 1391 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 𝜃 to represent the "external" assertibles. This is version a, which is without the phrase "or both"; see stoic4b 1392 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.)

((𝜑𝜓) → 𝜒)    &   ((𝜒𝜑𝜃) → 𝜏)       ((𝜑𝜓𝜃) → 𝜏)

Theoremstoic4b 1392 Stoic logic Thema 4 version b.

This is version b, which is with the phrase "or both". See stoic4a 1391 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.)

((𝜑𝜓) → 𝜒)    &   (((𝜒𝜑𝜓) ∧ 𝜃) → 𝜏)       ((𝜑𝜓𝜃) → 𝜏)

1.2.16  Logical implication (continued)

Theoremsyl6an 1393 A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))    &   ((𝜓𝜃) → 𝜏)       (𝜑 → (𝜒𝜏))

Theoremsyl10 1394 A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓 → (𝜃𝜏)))    &   (𝜒 → (𝜏𝜂))       (𝜑 → (𝜓 → (𝜃𝜂)))

Theoremexbir 1395 Exportation implication also converting head from biconditional to conditional. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓) → (𝜒𝜃)) → (𝜑 → (𝜓 → (𝜃𝜒))))

Theorem3impexp 1396 impexp 261 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))

Theorem3impexpbicom 1397 3impexp 1396 with biconditional consequent of antecedent that is commuted in consequent. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓𝜒) → (𝜃𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))))

Theorem3impexpbicomi 1398 Deduction form of 3impexpbicom 1397. (Contributed by Alan Sare, 31-Dec-2011.)
((𝜑𝜓𝜒) → (𝜃𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃))))

Theoremancomsimp 1399 Closed form of ancoms 266. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓) → 𝜒) ↔ ((𝜓𝜑) → 𝜒))

Theoremexpcomd 1400 Deduction form of expcom 115. (Contributed by Alan Sare, 22-Jul-2012.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜒 → (𝜓𝜃)))

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