P. 16 of Theorem List - New Foundations Explorer

Theorem List for New Foundations Explorer - 1501-1600   *Has distinct variable group(s)

Type	Label	Description
Statement
1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom
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 5, 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)
1.3.9  Derive the Lukasiewicz axioms from the The Russell-Bernays Axioms
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 5 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))
1.3.10  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 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 5, 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
1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes) 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: https://us.metamath.org/mpeuni/mmset.html#axiomnote 1542 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.
1.4.1  Universal quantifier; define "exists" and "not free"
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 2479 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (Ⅎxφ ↔ ∀x(φ → ∀xφ))
1.4.2  Rule scheme ax-gen (Generalization)
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
1.4.3  Axiom scheme ax-5 (Quantified Implication)
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 2457 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))