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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | merco2 1501 |
A single axiom for propositional calculus offered by Meredith.
This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1478. (Contributed by Anthony Hart, 7-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((φ → ψ) → (( ⊥ → χ) → θ)) → ((θ → φ) → (τ → (η → φ)))) | ||
Theorem | mercolem1 1502 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((φ → ψ) → χ) → (ψ → (θ → χ))) | ||
Theorem | mercolem2 1503 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((φ → ψ) → φ) → (χ → (θ → φ))) | ||
Theorem | mercolem3 1504 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((ψ → χ) → (ψ → (φ → χ))) | ||
Theorem | mercolem4 1505 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((θ → (η → φ)) → (((θ → χ) → φ) → (τ → (η → φ)))) | ||
Theorem | mercolem5 1506 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (θ → ((θ → φ) → (τ → (χ → φ)))) | ||
Theorem | mercolem6 1507 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((φ → (ψ → (φ → χ))) → (ψ → (φ → χ))) | ||
Theorem | mercolem7 1508 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((φ → ψ) → (((φ → χ) → (θ → ψ)) → (θ → ψ))) | ||
Theorem | mercolem8 1509 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((φ → ψ) → ((ψ → (φ → χ)) → (τ → (θ → (φ → χ))))) | ||
Theorem | re1tbw1 1510 | tbw-ax1 1465 rederived from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((φ → ψ) → ((ψ → χ) → (φ → χ))) | ||
Theorem | re1tbw2 1511 | tbw-ax2 1466 rederived from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (φ → (ψ → φ)) | ||
Theorem | re1tbw3 1512 | tbw-ax3 1467 rederived from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((φ → ψ) → φ) → φ) | ||
Theorem | re1tbw4 1513 |
tbw-ax4 1468 rederived from merco2 1501.
This theorem, along with re1tbw1 1510, re1tbw2 1511, and re1tbw3 1512, shows that merco2 1501, along with ax-mp 8, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ( ⊥ → φ) | ||
Theorem | rb-bijust 1514 | Justification for rb-imdf 1515. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((φ ↔ ψ) ↔ ¬ (¬ (¬ φ ∨ ψ) ∨ ¬ (¬ ψ ∨ φ))) | ||
Theorem | rb-imdf 1515 | The definition of implication, in terms of ∨ and ¬. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ (¬ (¬ (φ → ψ) ∨ (¬ φ ∨ ψ)) ∨ ¬ (¬ (¬ φ ∨ ψ) ∨ (φ → ψ))) | ||
Theorem | anmp 1516 | Modus ponens for ∨ ¬ axiom systems. (Contributed by Anthony Hart, 12-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ φ & ⊢ (¬ φ ∨ ψ) ⇒ ⊢ ψ | ||
Theorem | rb-ax1 1517 | The first of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ (¬ ψ ∨ χ) ∨ (¬ (φ ∨ ψ) ∨ (φ ∨ χ))) | ||
Theorem | rb-ax2 1518 | The second of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ (φ ∨ ψ) ∨ (ψ ∨ φ)) | ||
Theorem | rb-ax3 1519 | The third of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ φ ∨ (ψ ∨ φ)) | ||
Theorem | rb-ax4 1520 | The fourth of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ (φ ∨ φ) ∨ φ) | ||
Theorem | rbsyl 1521 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ ψ ∨ χ) & ⊢ (φ ∨ ψ) ⇒ ⊢ (φ ∨ χ) | ||
Theorem | rblem1 1522 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ φ ∨ ψ) & ⊢ (¬ χ ∨ θ) ⇒ ⊢ (¬ (φ ∨ χ) ∨ (ψ ∨ θ)) | ||
Theorem | rblem2 1523 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ (χ ∨ φ) ∨ (χ ∨ (φ ∨ ψ))) | ||
Theorem | rblem3 1524 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ (χ ∨ φ) ∨ ((χ ∨ ψ) ∨ φ)) | ||
Theorem | rblem4 1525 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ φ ∨ θ) & ⊢ (¬ ψ ∨ τ) & ⊢ (¬ χ ∨ η) ⇒ ⊢ (¬ ((φ ∨ ψ) ∨ χ) ∨ ((η ∨ τ) ∨ θ)) | ||
Theorem | rblem5 1526 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ (¬ ¬ φ ∨ ψ) ∨ (¬ ¬ ψ ∨ φ)) | ||
Theorem | rblem6 1527 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ (¬ (¬ φ ∨ ψ) ∨ ¬ (¬ ψ ∨ φ)) ⇒ ⊢ (¬ φ ∨ ψ) | ||
Theorem | rblem7 1528 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ (¬ (¬ φ ∨ ψ) ∨ ¬ (¬ ψ ∨ φ)) ⇒ ⊢ (¬ ψ ∨ φ) | ||
Theorem | re1axmp 1529 | ax-mp 8 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ φ & ⊢ (φ → ψ) ⇒ ⊢ ψ | ||
Theorem | re2luk1 1530 | luk-1 1420 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((φ → ψ) → ((ψ → χ) → (φ → χ))) | ||
Theorem | re2luk2 1531 | luk-2 1421 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((¬ φ → φ) → φ) | ||
Theorem | re2luk3 1532 |
luk-3 1422 derived from Russell-Bernays'.
This theorem, along with re1axmp 1529, re2luk1 1530, and re2luk2 1531 shows that rb-ax1 1517, rb-ax2 1518, rb-ax3 1519, and rb-ax4 1520, along with anmp 1516, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (φ → (¬ φ → ψ)) | ||
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 2301 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 8, modus tollendo tollens (modus tollens) mto 167, modus ponendo tollens I mpto1 1533, modus ponendo tollens II mpto2 1534, and modus tollendo ponens (exclusive-or version) mtp-xor 1536. 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 mtp-xor 1536 originally used exclusive-or, but over time the name modus tollendo ponens has increasingly referred to an inclusive-or variation, which is proved in mtp-or 1538. This set of indemonstrables is not the entire system of Stoic logic. | ||
Theorem | mpto1 1533 | 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 mpto2 1534) 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 & ⊢ ¬ (φ ∧ ψ) ⇒ ⊢ ¬ ψ | ||
Theorem | mpto2 1534 | 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, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 12-Nov-2017.) |
⊢ φ & ⊢ (φ ⊻ ψ) ⇒ ⊢ ¬ ψ | ||
Theorem | mpto2OLD 1535 | Obsolete version of mpto2 1534 as of 12-Nov-2017. (Contributed by David A. Wheeler, 3-Jul-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ φ & ⊢ (φ ⊻ ψ) ⇒ ⊢ ¬ ψ | ||
Theorem | mtp-xor 1536 | Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, 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 mtp-or 1538. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mpto2 1534, that is, it is exclusive-or df-xor 1305), 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 mpto2 1534), 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.) |
⊢ ¬ φ & ⊢ (φ ⊻ ψ) ⇒ ⊢ ψ | ||
Theorem | mtp-xorOLD 1537 | Obsolete version of mtp-xor 1536 as of 11-Nov-2017. (Contributed by David A. Wheeler, 4-Jul-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ¬ φ & ⊢ (φ ⊻ ψ) ⇒ ⊢ ψ | ||
Theorem | mtp-or 1538 | Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtp-xor 1536, 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 alternative 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 & ⊢ (φ ∨ ψ) ⇒ ⊢ ψ | ||
Theorem | mtp-orOLD 1539 | Obsolete version of mtp-or 1538 as of 11-Nov-2017. (Contributed by David A. Wheeler, 3-Jul-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ¬ φ & ⊢ (φ ∨ ψ) ⇒ ⊢ ψ | ||
Here we extend the language of wffs with predicate calculus, which allows us to talk about individual objects in a domain of discussion (which for us will be the universe of all sets, so we call them "setvar variables") and make true/false statements about predicates, which are relationships between objects, such as whether or not two objects are equal. In addition, we introduce universal quantification ("for all") in order to make statements about whether a wff holds for every object in the domain of discussion. Later we introduce existential quantification ("there exists", df-ex 1542) which is defined in terms of universal quantification. Our axioms are really axiom schemes, and our wff and setvar variables are metavariables ranging over expressions in an underlying "object language." This is explained here: http://us.metamath.org/mpeuni/mmset.html#axiomnote Our axiom system starts with the predicate calculus axiom schemes system S2 of Tarski defined in his 1965 paper, "A Simplified Formalization of Predicate Logic with Identity" [Tarski]. System S2 is defined in the last paragraph on p. 77, and repeated on p. 81 of [KalishMontague]. We do not include scheme B5 (our sp 1747) since [KalishMontague] shows it to be logically redundant (Lemma 9, p. 87, which we prove as theorem spw 1694 below). Theorem spw 1694 can be used to prove any instance of sp 1747 having no wff metavariables and mutually distinct setvar variables. However, it seems that sp 1747 in its general form cannot be derived from only Tarski's schemes. We do not include B5 i.e. sp 1747 as part of what we call "Tarski's system" because we want it to be the smallest set of axioms that is logically complete with no redundancies. We later prove sp 1747 as theorem ax4 2145 using the auxiliary axioms that make our system metalogically complete. Our version of Tarski's system S2 consists of propositional calculus plus ax-gen 1546, ax-5 1557, ax-17 1616, ax-9 1654, ax-8 1675, ax-13 1712, and ax-14 1714. The last 3 are equality axioms that represent 3 sub-schemes of Tarski's scheme B8. Due to its side-condition ("where φ is an atomic formula and ψ is obtained by replacing an occurrence of the variable x by the variable y"), we cannot represent his B8 directly without greatly complicating our scheme language, but the simpler schemes ax-8 1675, ax-13 1712, and ax-14 1714 are sufficient for set theory. Tarski's system is exactly equivalent to the traditional axiom system in most logic textbooks but has the advantage of being easy to manipulate with a computer program, and its simpler metalogic (with no built-in notions of free variable and proper substitution) is arguably easier for a non-logician human to follow step by step in a proof. However, in our system that derives schemes (rather than object language theorems) from other schemes, Tarski's S2 is not complete. For example, we cannot derive scheme sp 1747, even though (using spw 1694) we can derive all instances of it that don't involve wff metavariables or bundled setvar metavariables. (Two setvar metavariables are "bundled" if they can be substituted with the same setvar metavariable i.e. do not have a $d distinct variable proviso.) Later we will introduce auxiliary axiom schemes ax-6 1729, ax-7 1734, ax-12 1925, and ax-11 1746 that are metatheorems of Tarski's system (i.e. are logically redundant) but which give our system the property of "metalogical completeness," allowing us to prove directly (instead of, say, by induction on formula length) all possible schemes that can be expressed in our language. | ||
Syntax | wal 1540 | 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 | ||
Syntax | wex 1541 | Extend wff definition to include the existential quantifier ("there exists"). |
wff ∃xφ | ||
Definition | df-ex 1542 | Define existential quantification. ∃xφ means "there exists at least one set x such that φ is true." Definition of [Margaris] p. 49. (Contributed by NM, 5-Aug-1993.) |
⊢ (∃xφ ↔ ¬ ∀x ¬ φ) | ||
Theorem | alnex 1543 | Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀x ¬ φ ↔ ¬ ∃xφ) | ||
Syntax | wnf 1544 | Extend wff definition to include the not-free predicate. |
wff Ⅎxφ | ||
Definition | df-nf 1545 |
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 2026). An example of where this is used is
stdpc5 1798. See nf2 1866 for an alternative 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 (see nfequid 1678), 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). This predicate only applies to wffs. See df-nfc 2478 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (Ⅎxφ ↔ ∀x(φ → ∀xφ)) | ||
Axiom | ax-gen 1546 | Rule of Generalization. The postulated inference rule of pure 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 allt in set.mm shows the special case ∀x ⊤. Theorem spi 1753 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 | ||
Theorem | gen2 1547 | Generalization applied twice. (Contributed by NM, 30-Apr-1998.) |
⊢ φ ⇒ ⊢ ∀x∀yφ | ||
Theorem | mpg 1548 | Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.) |
⊢ (∀xφ → ψ) & ⊢ φ ⇒ ⊢ ψ | ||
Theorem | mpgbi 1549 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
⊢ (∀xφ ↔ ψ) & ⊢ φ ⇒ ⊢ ψ | ||
Theorem | mpgbir 1550 | 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 | ||
Theorem | nfi 1551 | Deduce that x is not free in φ from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (φ → ∀xφ) ⇒ ⊢ Ⅎxφ | ||
Theorem | hbth 1552 |
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) | ||
Theorem | nfth 1553 | No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ φ ⇒ ⊢ Ⅎxφ | ||
Theorem | nftru 1554 | The true constant has no free variables. (This can also be proven in one step with nfv 1619, but this proof does not use ax-17 1616.) (Contributed by Mario Carneiro, 6-Oct-2016.) |
⊢ Ⅎx ⊤ | ||
Theorem | nex 1555 | Generalization rule for negated wff. (Contributed by NM, 18-May-1994.) |
⊢ ¬ φ ⇒ ⊢ ¬ ∃xφ | ||
Theorem | nfnth 1556 | No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) |
⊢ ¬ φ ⇒ ⊢ Ⅎxφ | ||
Axiom | ax-5 1557 | Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ)) | ||
Theorem | alim 1558 | Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 30-Mar-2008.) |
⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ)) | ||
Theorem | alimi 1559 | Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ψ) ⇒ ⊢ (∀xφ → ∀xψ) | ||
Theorem | 2alimi 1560 | Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.) |
⊢ (φ → ψ) ⇒ ⊢ (∀x∀yφ → ∀x∀yψ) | ||
Theorem | al2imi 1561 | Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → (ψ → χ)) ⇒ ⊢ (∀xφ → (∀xψ → ∀xχ)) | ||
Theorem | alanimi 1562 | Variant of al2imi 1561 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.) |
⊢ ((φ ∧ ψ) → χ) ⇒ ⊢ ((∀xφ ∧ ∀xψ) → ∀xχ) | ||
Theorem | alimdh 1563 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 4-Jan-2002.) |
⊢ (φ → ∀xφ) & ⊢ (φ → (ψ → χ)) ⇒ ⊢ (φ → (∀xψ → ∀xχ)) | ||
Theorem | albi 1564 | Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀x(φ ↔ ψ) → (∀xφ ↔ ∀xψ)) | ||
Theorem | alrimih 1565 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ∀xφ) & ⊢ (φ → ψ) ⇒ ⊢ (φ → ∀xψ) | ||
Theorem | albii 1566 | Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.) |
⊢ (φ ↔ ψ) ⇒ ⊢ (∀xφ ↔ ∀xψ) | ||
Theorem | 2albii 1567 | Inference adding two universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.) |
⊢ (φ ↔ ψ) ⇒ ⊢ (∀x∀yφ ↔ ∀x∀yψ) | ||
Theorem | hbxfrbi 1568 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2456 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (φ ↔ ψ) & ⊢ (ψ → ∀xψ) ⇒ ⊢ (φ → ∀xφ) | ||
Theorem | nfbii 1569 | Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (φ ↔ ψ) ⇒ ⊢ (Ⅎxφ ↔ Ⅎxψ) | ||
Theorem | nfxfr 1570 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (φ ↔ ψ) & ⊢ Ⅎxψ ⇒ ⊢ Ⅎxφ | ||
Theorem | nfxfrd 1571 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ (φ ↔ ψ) & ⊢ (χ → Ⅎxψ) ⇒ ⊢ (χ → Ⅎxφ) | ||
Theorem | alex 1572 | Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀xφ ↔ ¬ ∃x ¬ φ) | ||
Theorem | 2nalexn 1573 | Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (¬ ∀x∀yφ ↔ ∃x∃y ¬ φ) | ||
Theorem | exnal 1574 | Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (∃x ¬ φ ↔ ¬ ∀xφ) | ||
Theorem | exim 1575 | Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) |
⊢ (∀x(φ → ψ) → (∃xφ → ∃xψ)) | ||
Theorem | eximi 1576 | Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ψ) ⇒ ⊢ (∃xφ → ∃xψ) | ||
Theorem | 2eximi 1577 | Inference adding two existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.) |
⊢ (φ → ψ) ⇒ ⊢ (∃x∃yφ → ∃x∃yψ) | ||
Theorem | alinexa 1578 | A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.) |
⊢ (∀x(φ → ¬ ψ) ↔ ¬ ∃x(φ ∧ ψ)) | ||
Theorem | alexn 1579 | A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.) |
⊢ (∀x∃y ¬ φ ↔ ¬ ∃x∀yφ) | ||
Theorem | 2exnexn 1580 | Theorem *11.51 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) (Proof shortened by Wolf Lammen, 25-Sep-2014.) |
⊢ (∃x∀yφ ↔ ¬ ∀x∃y ¬ φ) | ||
Theorem | exbi 1581 | Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀x(φ ↔ ψ) → (∃xφ ↔ ∃xψ)) | ||
Theorem | exbii 1582 | Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.) |
⊢ (φ ↔ ψ) ⇒ ⊢ (∃xφ ↔ ∃xψ) | ||
Theorem | 2exbii 1583 | Inference adding two existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.) |
⊢ (φ ↔ ψ) ⇒ ⊢ (∃x∃yφ ↔ ∃x∃yψ) | ||
Theorem | 3exbii 1584 | Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.) |
⊢ (φ ↔ ψ) ⇒ ⊢ (∃x∃y∃zφ ↔ ∃x∃y∃zψ) | ||
Theorem | exanali 1585 | A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996.) (Proof shortened by Wolf Lammen, 4-Sep-2014.) |
⊢ (∃x(φ ∧ ¬ ψ) ↔ ¬ ∀x(φ → ψ)) | ||
Theorem | exancom 1586 | Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.) |
⊢ (∃x(φ ∧ ψ) ↔ ∃x(ψ ∧ φ)) | ||
Theorem | alrimdh 1587 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (φ → ∀xφ) & ⊢ (ψ → ∀xψ) & ⊢ (φ → (ψ → χ)) ⇒ ⊢ (φ → (ψ → ∀xχ)) | ||
Theorem | eximdh 1588 | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.) |
⊢ (φ → ∀xφ) & ⊢ (φ → (ψ → χ)) ⇒ ⊢ (φ → (∃xψ → ∃xχ)) | ||
Theorem | nexdh 1589 | Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.) |
⊢ (φ → ∀xφ) & ⊢ (φ → ¬ ψ) ⇒ ⊢ (φ → ¬ ∃xψ) | ||
Theorem | albidh 1590 | Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ∀xφ) & ⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∀xψ ↔ ∀xχ)) | ||
Theorem | exbidh 1591 | Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ∀xφ) & ⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∃xψ ↔ ∃xχ)) | ||
Theorem | exsimpl 1592 | Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ (∃x(φ ∧ ψ) → ∃xφ) | ||
Theorem | 19.26 1593 | Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 147. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) |
⊢ (∀x(φ ∧ ψ) ↔ (∀xφ ∧ ∀xψ)) | ||
Theorem | 19.26-2 1594 | Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.) |
⊢ (∀x∀y(φ ∧ ψ) ↔ (∀x∀yφ ∧ ∀x∀yψ)) | ||
Theorem | 19.26-3an 1595 | Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.) |
⊢ (∀x(φ ∧ ψ ∧ χ) ↔ (∀xφ ∧ ∀xψ ∧ ∀xχ)) | ||
Theorem | 19.29 1596 | Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((∀xφ ∧ ∃xψ) → ∃x(φ ∧ ψ)) | ||
Theorem | 19.29r 1597 | Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.) |
⊢ ((∃xφ ∧ ∀xψ) → ∃x(φ ∧ ψ)) | ||
Theorem | 19.29r2 1598 | Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.) |
⊢ ((∃x∃yφ ∧ ∀x∀yψ) → ∃x∃y(φ ∧ ψ)) | ||
Theorem | 19.29x 1599 | Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.) |
⊢ ((∃x∀yφ ∧ ∀x∃yψ) → ∃x∃y(φ ∧ ψ)) | ||
Theorem | 19.35 1600 | Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.) |
⊢ (∃x(φ → ψ) ↔ (∀xφ → ∃xψ)) |
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