P. 15 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 1401-1500   *Has distinct variable group(s)

Type	Label	Description
Statement
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 104, ∨ (disjunction aka logical inclusive 'or') wo 708, → (implies) wi 4, ¬ (not) wn 3, ↔ (logical equivalence) df-bi 117, and ⊻ (exclusive or) df-xor 1376.
Theorem	truantru 1401	A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊤ ∧ ⊤) ↔ ⊤)
Theorem	truanfal 1402	A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊤ ∧ ⊥) ↔ ⊥)
Theorem	falantru 1403	A ∧ identity. (Contributed by David A. Wheeler, 23-Feb-2018.)
⊢ ((⊥ ∧ ⊤) ↔ ⊥)
Theorem	falanfal 1404	A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊥ ∧ ⊥) ↔ ⊥)
Theorem	truortru 1405	A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊤ ∨ ⊤) ↔ ⊤)
Theorem	truorfal 1406	A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊤ ∨ ⊥) ↔ ⊤)
Theorem	falortru 1407	A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊥ ∨ ⊤) ↔ ⊤)
Theorem	falorfal 1408	A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊥ ∨ ⊥) ↔ ⊥)
Theorem	truimtru 1409	A → identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊤ → ⊤) ↔ ⊤)
Theorem	truimfal 1410	A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊤ → ⊥) ↔ ⊥)
Theorem	falimtru 1411	A → identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊥ → ⊤) ↔ ⊤)
Theorem	falimfal 1412	A → identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊥ → ⊥) ↔ ⊤)
Theorem	nottru 1413	A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ (¬ ⊤ ↔ ⊥)
Theorem	notfal 1414	A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ (¬ ⊥ ↔ ⊤)
Theorem	trubitru 1415	A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊤ ↔ ⊤) ↔ ⊤)
Theorem	trubifal 1416	A ↔ identity. (Contributed by David A. Wheeler, 23-Feb-2018.)
⊢ ((⊤ ↔ ⊥) ↔ ⊥)
Theorem	falbitru 1417	A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊥ ↔ ⊤) ↔ ⊥)
Theorem	falbifal 1418	A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊥ ↔ ⊥) ↔ ⊤)
Theorem	truxortru 1419	A ⊻ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
⊢ ((⊤ ⊻ ⊤) ↔ ⊥)
Theorem	truxorfal 1420	A ⊻ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
⊢ ((⊤ ⊻ ⊥) ↔ ⊤)
Theorem	falxortru 1421	A ⊻ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
⊢ ((⊥ ⊻ ⊤) ↔ ⊤)
Theorem	falxorfal 1422	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 662, modus ponendo tollens I mptnan 1423, modus ponendo tollens II mptxor 1424, and modus tollendo ponens (exclusive-or version) mtpxor 1426. 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 1426 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 1425. This set of indemonstrables is not the entire system of Stoic logic.
Theorem	mptnan 1423	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 1424) 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	mptxor 1424	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    &   ⊢ (𝜑 ⊻ 𝜓)    ⇒   ⊢ ¬ 𝜓
Theorem	mtpor 1425	Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1426, 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    &   ⊢ (𝜑 ∨ 𝜓)    ⇒   ⊢ 𝜓
Theorem	mtpxor 1426	Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1425, 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 1425. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1424, that is, it is exclusive-or df-xor 1376), 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 1424), 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    &   ⊢ (𝜑 ⊻ 𝜓)    ⇒   ⊢ 𝜓
Theorem	stoic1a 1427	Stoic logic Thema 1 (part a). The first thema of the four Stoic logic themata, in its basic form, was: "When from two (assertibles) a third follows, then from either of them together with the contradictory of the conclusion the contradictory of the other follows." (Apuleius Int. 209.9-14), see [Bobzien] p. 117 and https://plato.stanford.edu/entries/logic-ancient/ We will represent thema 1 as two very similar rules stoic1a 1427 and stoic1b 1428 to represent each side. (Contributed by David A. Wheeler, 16-Feb-2019.) (Proof shortened by Wolf Lammen, 21-May-2020.)
⊢ ((𝜑 ∧ 𝜓) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ ¬ 𝜃) → ¬ 𝜓)
Theorem	stoic1b 1428	Stoic logic Thema 1 (part b). The other part of thema 1 of Stoic logic; see stoic1a 1427. (Contributed by David A. Wheeler, 16-Feb-2019.)
⊢ ((𝜑 ∧ 𝜓) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ ¬ 𝜃) → ¬ 𝜑)
Theorem	stoic2a 1429	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, 𝜓⊢ 𝜒; in Metamath we will represent that construct as 𝜑 ∧ 𝜓 → 𝜒. This version a is without the phrase "or both"; see stoic2b 1430 for the version with the phrase "or both". We already have this rule as syldan 282, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	stoic2b 1430	Stoic logic Thema 2 version b. See stoic2a 1429. Version b is with the phrase "or both". We already have this rule as mpd3an3 1338, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	stoic3 1431	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.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)
Theorem	stoic4a 1432	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 1433 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)
Theorem	stoic4b 1433	Stoic logic Thema 4 version b. This is version b, which is with the phrase "or both". See stoic4a 1432 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ (((𝜒 ∧ 𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)
1.2.16  Logical implication (continued)
Theorem	syl6an 1434	A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜒 → 𝜃))    &   ⊢ ((𝜓 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜑 → (𝜒 → 𝜏))
Theorem	syl10 1435	A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))    &   ⊢ (𝜒 → (𝜏 → 𝜂))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂)))
Theorem	exbir 1436	Exportation implication also converting head from biconditional to conditional. (Contributed by Alan Sare, 31-Dec-2011.)
⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜃 → 𝜒))))
Theorem	3impexp 1437	impexp 263 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))
Theorem	3impexpbicom 1438	3impexp 1437 with biconditional consequent of antecedent that is commuted in consequent. (Contributed by Alan Sare, 31-Dec-2011.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃)))))
Theorem	3impexpbicomi 1439	Deduction form of 3impexpbicom 1438. (Contributed by Alan Sare, 31-Dec-2011.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 ↔ 𝜃))))
Theorem	ancomsimp 1440	Closed form of ancoms 268. (Contributed by Alan Sare, 31-Dec-2011.)
⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜓 ∧ 𝜑) → 𝜒))
Theorem	expcomd 1441	Deduction form of expcom 116. (Contributed by Alan Sare, 22-Jul-2012.)
⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))    ⇒   ⊢ (𝜑 → (𝜒 → (𝜓 → 𝜃)))
Theorem	expdcom 1442	Commuted form of expd 258. (Contributed by Alan Sare, 18-Mar-2012.)
⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))    ⇒   ⊢ (𝜓 → (𝜒 → (𝜑 → 𝜃)))
Theorem	simplbi2comg 1443	Implication form of simplbi2com 1444. (Contributed by Alan Sare, 22-Jul-2012.)
⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓 → 𝜑)))
Theorem	simplbi2com 1444	A deduction eliminating a conjunct, similar to simplbi2 385. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜒 → (𝜓 → 𝜑))
Theorem	syl6ci 1445	A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜒 → (𝜃 → 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜏))
Theorem	mpisyl 1446	A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.)
⊢ (𝜑 → 𝜓)    &   ⊢ 𝜒    &   ⊢ (𝜓 → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → 𝜃)
1.3  Predicate calculus mostly without distinct variables
1.3.1  Universal quantifier (continued) The universal quantifier was introduced above in wal 1351 for use by df-tru 1356. See the comments in that section. In this section, we continue with the first "real" use of it.
Axiom	ax-5 1447	Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Axiom	ax-7 1448	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.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Axiom	ax-gen 1449	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 1536 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 1450	Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
⊢ 𝜑    ⇒   ⊢ ∀𝑥∀𝑦𝜑
Theorem	mpg 1451	Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
⊢ (∀𝑥𝜑 → 𝜓)    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	mpgbi 1452	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 1453	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	a7s 1454	Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥∀𝑦𝜑 → 𝜓)    ⇒   ⊢ (∀𝑦∀𝑥𝜑 → 𝜓)
Theorem	alimi 1455	Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑥𝜓)
Theorem	2alimi 1456	Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)
Theorem	alim 1457	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	al2imi 1458	Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
Theorem	alanimi 1459	Variant of al2imi 1458 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)
Syntax	wnf 1460	Extend wff definition to include the not-free predicate.
wff Ⅎ𝑥𝜑
Definition	df-nf 1461	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 1777). An example of where this is used is stdpc5 1584. See nf2 1668 for an alternate definition which does not involve nested quantifiers on the same variable. Nonfreeness is a commonly used condition, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the notion of nonfreeness within the logic itself rather than as a metalogical side condition. To be precise, our definition really means "effectively not free", because it is slightly less restrictive than the usual textbook definition for "not free" (which considers syntactic freedom). For example, 𝑥 is effectively not free in the expression 𝑥 = 𝑥 (even though 𝑥 is syntactically free in it, so would be considered "free" in the usual textbook definition) because the value of 𝑥 in the formula 𝑥 = 𝑥 does not affect the truth of that formula (and thus substitutions will not change the result), see nfequid 1702. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
Theorem	nfi 1462	Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	hbth 1463	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 1464	No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ 𝜑    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	nfnth 1465	No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.)
⊢ ¬ 𝜑    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	nftru 1466	The true constant has no free variables. (This can also be proven in one step with nfv 1528, but this proof does not use ax-17 1526.) (Contributed by Mario Carneiro, 6-Oct-2016.)
⊢ Ⅎ𝑥⊤
Theorem	alimdh 1467	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 1468	Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
Theorem	alrimih 1469	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜓)
Theorem	albii 1470	Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
Theorem	2albii 1471	Inference adding 2 universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓)
Theorem	hbxfrbi 1472	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜑)
Theorem	nfbii 1473	Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)
Theorem	nfxfr 1474	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 1475	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	alcoms 1476	Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.)
⊢ (∀𝑥∀𝑦𝜑 → 𝜓)    ⇒   ⊢ (∀𝑦∀𝑥𝜑 → 𝜓)
Theorem	hbal 1477	If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)
Theorem	alcom 1478	Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑦∀𝑥𝜑)
Theorem	alrimdh 1479	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	albidh 1480	Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Theorem	19.26 1481	Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
Theorem	19.26-2 1482	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 1483	Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
⊢ (∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓 ∧ ∀𝑥𝜒))
Theorem	19.33 1484	Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓))
Theorem	alrot3 1485	Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦∀𝑧𝜑 ↔ ∀𝑦∀𝑧∀𝑥𝜑)
Theorem	alrot4 1486	Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 28-Jun-2014.)
⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑)
Theorem	albiim 1487	Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑)))
Theorem	2albiim 1488	Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) ↔ (∀𝑥∀𝑦(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦(𝜓 → 𝜑)))
Theorem	hband 1489	Deduction form of bound-variable hypothesis builder hban 1547. (Contributed by NM, 2-Jan-2002.)
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜒)))
Theorem	hb3and 1490	Deduction form of bound-variable hypothesis builder hb3an 1550. (Contributed by NM, 17-Feb-2013.)
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    &   ⊢ (𝜑 → (𝜃 → ∀𝑥𝜃))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → ∀𝑥(𝜓 ∧ 𝜒 ∧ 𝜃)))
Theorem	hbald 1491	Deduction form of bound-variable hypothesis builder hbal 1477. (Contributed by NM, 2-Jan-2002.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓))
Syntax	wex 1492	Extend wff definition to include the existential quantifier ("there exists").
wff ∃𝑥𝜑
Axiom	ax-ie1 1493	𝑥 is bound in ∃𝑥𝜑. One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
Axiom	ax-ie2 1494	Define existential quantification. ∃𝑥𝜑 means "there exists at least one set 𝑥 such that 𝜑 is true". One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
Theorem	hbe1 1495	𝑥 is not free in ∃𝑥𝜑. (Contributed by NM, 5-Aug-1993.)
⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
Theorem	nfe1 1496	𝑥 is not free in ∃𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥∃𝑥𝜑
Theorem	19.23ht 1497	Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 1-Feb-2015.)
⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
Theorem	19.23h 1498	Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.)
⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
Theorem	alnex 1499	Theorem 19.7 of [Margaris] p. 89. To read this intuitionistically, think of it as "if 𝜑 can be refuted for all 𝑥, then it is not possible to find an 𝑥 for which 𝜑 holds" (and likewise for the converse). Comparing this with dfexdc 1501 illustrates that statements which look similar (to someone used to classical logic) can be different intuitionistically due to different placement of negations. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 1-Feb-2015.) (Revised by Mario Carneiro, 12-May-2015.)
⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
Theorem	nex 1500	Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
⊢ ¬ 𝜑    ⇒   ⊢ ¬ ∃𝑥𝜑