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

Theorema1tru 1301 Anything implies . (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
(𝜑 → ⊤)

Theoremtruan 1302 True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
((⊤ ∧ 𝜑) ↔ 𝜑)

Theoremdfnot 1303 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 1304 Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑𝜓) → ⊥)       (𝜑 → ¬ 𝜓)

Theorempm2.21fal 1305 If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜓)       (𝜑 → ⊥)

Theorempclem6 1306 Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof rewritten by Jim Kingdon, 4-May-2018.)
((𝜑 ↔ (𝜓 ∧ ¬ 𝜑)) → ¬ 𝜓)

1.2.14  Logical 'xor'

Syntaxwxo 1307 Extend wff definition to include exclusive disjunction ('xor').
wff (𝜑𝜓)

Definitiondf-xor 1308 Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. Contrast with (wa 102), (wo 662), and (wi 4) . (Contributed by FL, 22-Nov-2010.) (Modified by Jim Kingdon, 1-Mar-2018.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))

Theoremxoranor 1309 One way of defining exclusive or. Equivalent to df-xor 1308. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)))

Theoremexcxor 1310 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 1311 XOR implies OR. (Contributed by BJ, 19-Apr-2019.)
((𝜑𝜓) → (𝜑𝜓))

Theoremxorbi2d 1312 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 1313 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 1314 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 1315 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))

Theoremxorbin 1316 A consequence of exclusive or. In classical logic the converse also holds. (Contributed by Jim Kingdon, 8-Mar-2018.)
((𝜑𝜓) → (𝜑 ↔ ¬ 𝜓))

Theorempm5.18im 1317 One direction of pm5.18dc 811, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.)
((𝜑𝜓) → ¬ (𝜑 ↔ ¬ 𝜓))

Theoremxornbi 1318 A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1323. (Contributed by Jim Kingdon, 10-Mar-2018.)
((𝜑𝜓) → ¬ (𝜑𝜓))

Theoremxor3dc 1319 Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))

Theoremxorcom 1320 is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.)
((𝜑𝜓) ↔ (𝜓𝜑))

Theorempm5.15dc 1321 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 1322 Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))))

Theoremxornbidc 1323 Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑𝜓))))

Theoremxordc 1324 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 1325 Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.)
((𝜑𝜓) → DECID 𝜑)

Theoremnbbndc 1326 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 1327 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 1328 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 1329 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 1330 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 1331 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 1332 Conjunction distributes over exclusive-or. (Contributed by Mario Carneiro and Jim Kingdon, 7-Oct-2018.)
((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒)))

1.2.15  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 102, (disjunction aka logical inclusive 'or') wo 662, (implies) wi 4, ¬ (not) wn 3, (logical equivalence) df-bi 115, and (exclusive or) df-xor 1308.

Theoremtruantru 1333 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∧ ⊤) ↔ ⊤)

Theoremtruanfal 1334 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∧ ⊥) ↔ ⊥)

Theoremfalantru 1335 A identity. (Contributed by David A. Wheeler, 23-Feb-2018.)
((⊥ ∧ ⊤) ↔ ⊥)

Theoremfalanfal 1336 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∧ ⊥) ↔ ⊥)

Theoremtruortru 1337 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ∨ ⊤) ↔ ⊤)

Theoremtruorfal 1338 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∨ ⊥) ↔ ⊤)

Theoremfalortru 1339 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∨ ⊤) ↔ ⊤)

Theoremfalorfal 1340 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ∨ ⊥) ↔ ⊥)

Theoremtruimtru 1341 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ → ⊤) ↔ ⊤)

Theoremtruimfal 1342 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ → ⊥) ↔ ⊥)

Theoremfalimtru 1343 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ → ⊤) ↔ ⊤)

Theoremfalimfal 1344 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ → ⊥) ↔ ⊤)

Theoremnottru 1345 A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(¬ ⊤ ↔ ⊥)

Theoremnotfal 1346 A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(¬ ⊥ ↔ ⊤)

Theoremtrubitru 1347 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ↔ ⊤) ↔ ⊤)

Theoremtrubifal 1348 A identity. (Contributed by David A. Wheeler, 23-Feb-2018.)
((⊤ ↔ ⊥) ↔ ⊥)

Theoremfalbitru 1349 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ↔ ⊤) ↔ ⊥)

Theoremfalbifal 1350 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ↔ ⊥) ↔ ⊤)

Theoremtruxortru 1351 A identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
((⊤ ⊻ ⊤) ↔ ⊥)

Theoremtruxorfal 1352 A identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
((⊤ ⊻ ⊥) ↔ ⊤)

Theoremfalxortru 1353 A identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
((⊥ ⊻ ⊤) ↔ ⊤)

Theoremfalxorfal 1354 A identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
((⊥ ⊻ ⊥) ↔ ⊥)

1.2.16  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 7, modus tollendo tollens (modus tollens) mto 621, modus ponendo tollens I mptnan 1355, modus ponendo tollens II mptxor 1356, and modus tollendo ponens (exclusive-or version) mtpxor 1358. 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 1358 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 1357. This set of indemonstrables is not the entire system of Stoic logic.

Theoremmptnan 1355 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 1356) 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 1356 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 1357 Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1358, 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 1358 Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1357, 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 1357. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1356, that is, it is exclusive-or df-xor 1308), 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 1356), 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 1359 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 1360 for the version with the phrase "or both". We already have this rule as syldan 276, 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 1360 Stoic logic Thema 2 version b. See stoic2a 1359.

Version b is with the phrase "or both". We already have this rule as mpd3an3 1270, 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 1361 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 1362 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 1363 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.)

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

Theoremstoic4b 1363 Stoic logic Thema 4 version b.

This is version b, which is with the phrase "or both". See stoic4a 1362 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.)

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

1.2.17  Logical implication (continued)

Theoremsyl6an 1364 A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))    &   ((𝜓𝜃) → 𝜏)       (𝜑 → (𝜒𝜏))

Theoremsyl10 1365 A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓 → (𝜃𝜏)))    &   (𝜒 → (𝜏𝜂))       (𝜑 → (𝜓 → (𝜃𝜂)))

Theoremexbir 1366 Exportation implication also converting head from biconditional to conditional. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓) → (𝜒𝜃)) → (𝜑 → (𝜓 → (𝜃𝜒))))

Theorem3impexp 1367 impexp 259 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))

Theorem3impexpbicom 1368 3impexp 1367 with biconditional consequent of antecedent that is commuted in consequent. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓𝜒) → (𝜃𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))))

Theorem3impexpbicomi 1369 Deduction form of 3impexpbicom 1368. (Contributed by Alan Sare, 31-Dec-2011.)
((𝜑𝜓𝜒) → (𝜃𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃))))

Theoremancomsimp 1370 Closed form of ancoms 264. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓) → 𝜒) ↔ ((𝜓𝜑) → 𝜒))

Theoremexpcomd 1371 Deduction form of expcom 114. (Contributed by Alan Sare, 22-Jul-2012.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜑 → (𝜒 → (𝜓𝜃)))

Theoremexpdcom 1372 Commuted form of expd 254. (Contributed by Alan Sare, 18-Mar-2012.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (𝜓 → (𝜒 → (𝜑𝜃)))

Theoremsimplbi2comg 1373 Implication form of simplbi2com 1374. (Contributed by Alan Sare, 22-Jul-2012.)
((𝜑 ↔ (𝜓𝜒)) → (𝜒 → (𝜓𝜑)))

Theoremsimplbi2com 1374 A deduction eliminating a conjunct, similar to simplbi2 377. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.)
(𝜑 ↔ (𝜓𝜒))       (𝜒 → (𝜓𝜑))

Theoremsyl6ci 1375 A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)    &   (𝜒 → (𝜃𝜏))       (𝜑 → (𝜓𝜏))

Theoremmpisyl 1376 A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.)
(𝜑𝜓)    &   𝜒    &   (𝜓 → (𝜒𝜃))       (𝜑𝜃)

1.3  Predicate calculus mostly without distinct variables

1.3.1  Universal quantifier (continued)

The universal quantifier was introduced above in wal 1283 for use by df-tru 1288. See the comments in that section. In this section, we continue with the first "real" use of it.

Axiomax-5 1377 Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Axiomax-7 1378 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.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Axiomax-gen 1379 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 𝑥 = 𝑥, we can conclude 𝑥𝑥 = 𝑥 or even 𝑦𝑥 = 𝑥. Theorem spi 1470 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.)
𝜑       𝑥𝜑

Theoremgen2 1380 Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
𝜑       𝑥𝑦𝜑

Theoremmpg 1381 Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
(∀𝑥𝜑𝜓)    &   𝜑       𝜓

Theoremmpgbi 1382 Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
(∀𝑥𝜑𝜓)    &   𝜑       𝜓

Theoremmpgbir 1383 Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
(𝜑 ↔ ∀𝑥𝜓)    &   𝜓       𝜑

Theorema7s 1384 Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.)
(∀𝑥𝑦𝜑𝜓)       (∀𝑦𝑥𝜑𝜓)

Theoremalimi 1385 Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       (∀𝑥𝜑 → ∀𝑥𝜓)

Theorem2alimi 1386 Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
(𝜑𝜓)       (∀𝑥𝑦𝜑 → ∀𝑥𝑦𝜓)

Theoremalim 1387 Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 30-Mar-2008.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Theoremal2imi 1388 Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Theoremalanimi 1389 Variant of al2imi 1388 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
((𝜑𝜓) → 𝜒)       ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Syntaxwnf 1390 Extend wff definition to include the not-free predicate.
wff 𝑥𝜑

Definitiondf-nf 1391 Define the not-free predicate for wffs. This is read "𝑥 is not free in 𝜑". Not-free means that the value of 𝑥 cannot affect the value of 𝜑, e.g., any occurrence of 𝑥 in 𝜑 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 1702). An example of where this is used is stdpc5 1517. See nf2 1599 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, 𝑥 is effectively not free in the bare expression 𝑥 = 𝑥, even though 𝑥 would be considered free in the usual textbook definition, because the value of 𝑥 in the expression 𝑥 = 𝑥 cannot affect the truth of the expression (and thus substitution will not change the result). (Contributed by Mario Carneiro, 11-Aug-2016.)

(Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Theoremnfi 1392 Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑 → ∀𝑥𝜑)       𝑥𝜑

Theoremhbth 1393 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 (𝜑 → ∀𝑥𝜑) from smaller formulas of this form. These are useful for constructing hypotheses that state "𝑥 is (effectively) not free in 𝜑." (Contributed by NM, 5-Aug-1993.)

𝜑       (𝜑 → ∀𝑥𝜑)

Theoremnfth 1394 No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝜑       𝑥𝜑

Theoremnfnth 1395 No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.)
¬ 𝜑       𝑥𝜑

Theoremnftru 1396 The true constant has no free variables. (This can also be proven in one step with nfv 1462, but this proof does not use ax-17 1460.) (Contributed by Mario Carneiro, 6-Oct-2016.)
𝑥

Theoremalimdh 1397 Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 4-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Theoremalbi 1398 Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))

Theoremalrimih 1399 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑𝜓)       (𝜑 → ∀𝑥𝜓)

Theoremalbii 1400 Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.)
(𝜑𝜓)       (∀𝑥𝜑 ↔ ∀𝑥𝜓)

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