P. 16 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 1501-1600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nanan 1501	Conjunction in terms of alternative denial. (Contributed by Mario Carneiro, 9-May-2015.)
⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ⊼ 𝜓))
Theorem	dfnan2 1502	Alternative denial in terms of our primitive connectives (implication and negation). (Contributed by WL, 26-Jun-2020.)
⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜑 → ¬ 𝜓))
Theorem	nanor 1503	Alternative denial in terms of disjunction and negation. This explains the name "alternative denial". (Contributed by BJ, 19-Oct-2022.)
⊢ ((𝜑 ⊼ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
Theorem	nancom 1504	Alternative denial is commutative. Remark: alternative denial is not associative, see nanass 1518. (Contributed by Mario Carneiro, 9-May-2015.) (Proof shortened by Wolf Lammen, 26-Jun-2020.)
⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜓 ⊼ 𝜑))
Theorem	nannan 1505	Nested alternative denials. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 26-Jun-2020.)
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒)))
Theorem	nanim 1506	Implication in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007.)
⊢ ((𝜑 → 𝜓) ↔ (𝜑 ⊼ (𝜓 ⊼ 𝜓)))
Theorem	nannot 1507	Negation in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007.) Use dfnan2 1502. (Revised by Wolf Lammen, 26-Jun-2020.)
⊢ (¬ 𝜑 ↔ (𝜑 ⊼ 𝜑))
Theorem	nanbi 1508	Biconditional in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 27-Jun-2020.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓))))
Theorem	nanbi1 1509	Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by Anthony Hart, 1-Sep-2011.) (Proof shortened by Wolf Lammen, 27-Jun-2020.)
⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜒)))
Theorem	nanbi2 1510	Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by Anthony Hart, 1-Sep-2011.) (Proof shortened by SF, 2-Jan-2018.)
⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ⊼ 𝜑) ↔ (𝜒 ⊼ 𝜓)))
Theorem	nanbi12 1511	Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜃)))
Theorem	nanbi1i 1512	Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜒))
Theorem	nanbi2i 1513	Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜒 ⊼ 𝜑) ↔ (𝜒 ⊼ 𝜓))
Theorem	nanbi12i 1514	Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜃))
Theorem	nanbi1d 1515	Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜓 ⊼ 𝜃) ↔ (𝜒 ⊼ 𝜃)))
Theorem	nanbi2d 1516	Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜃 ⊼ 𝜓) ↔ (𝜃 ⊼ 𝜒)))
Theorem	nanbi12d 1517	Join two logical equivalences with anti-conjunction. (Contributed by Scott Fenton, 2-Jan-2018.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 ⊼ 𝜃) ↔ (𝜒 ⊼ 𝜏)))
Theorem	nanass 1518	A characterization of when an expression involving alternative denials associates. Remark: alternative denial is commutative, see nancom 1504. (Contributed by Richard Penner, 29-Feb-2020.) (Proof shortened by Wolf Lammen, 23-Oct-2022.)
⊢ ((𝜑 ↔ 𝜒) ↔ (((𝜑 ⊼ 𝜓) ⊼ 𝜒) ↔ (𝜑 ⊼ (𝜓 ⊼ 𝜒))))
1.2.13  Logical "xor"
Syntax	wxo 1519	Extend wff definition to include exclusive disjunction ("xor").
wff (𝜑 ⊻ 𝜓)
Definition	df-xor 1520	Define exclusive disjunction (logical "xor"). Return true if either the left or right, but not both, are true. After we define the constant true ⊤ (df-tru 1551) and the constant false ⊥ (df-fal 1561), we will be able to prove these truth table values: ((⊤ ⊻ ⊤) ↔ ⊥) (truxortru 1593), ((⊤ ⊻ ⊥) ↔ ⊤) (truxorfal 1594), ((⊥ ⊻ ⊤) ↔ ⊤) (falxortru 1595), and ((⊥ ⊻ ⊥) ↔ ⊥) (falxorfal 1596). Contrast with ∧ (df-an 398), ∨ (df-or 855), → (wi 4), and ⊼ (df-nan 1500). (Contributed by FL, 22-Nov-2010.)
⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
Theorem	xnor 1521	Two ways to write XNOR (exclusive not-or). (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓))
Theorem	xorcom 1522	The connector ⊻ is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 21-Apr-2024.)
⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑))
Theorem	xorass 1523	The connector ⊻ is associative. (Contributed by FL, 22-Nov-2010.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Wolf Lammen, 20-Jun-2020.)
⊢ (((𝜑 ⊻ 𝜓) ⊻ 𝜒) ↔ (𝜑 ⊻ (𝜓 ⊻ 𝜒)))
Theorem	excxor 1524	This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.)
⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓)))
Theorem	xor2 1525	Two ways to express "exclusive or". (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
Theorem	xoror 1526	Exclusive disjunction implies disjunction ("XOR implies OR"). (Contributed by BJ, 19-Apr-2019.)
⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ∨ 𝜓))
Theorem	xornan 1527	Exclusive disjunction implies alternative denial ("XOR implies NAND"). (Contributed by BJ, 19-Apr-2019.)
⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ∧ 𝜓))
Theorem	xornan2 1528	XOR implies NAND (written with the ⊼ connector). (Contributed by BJ, 19-Apr-2019.)
⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ⊼ 𝜓))
Theorem	xorneg2 1529	The connector ⊻ is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 27-Jun-2020.)
⊢ ((𝜑 ⊻ ¬ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓))
Theorem	xorneg1 1530	The connector ⊻ is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 27-Jun-2020.)
⊢ ((¬ 𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓))
Theorem	xorneg 1531	The connector ⊻ is unchanged under negation of both arguments. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ ((¬ 𝜑 ⊻ ¬ 𝜓) ↔ (𝜑 ⊻ 𝜓))
Theorem	xorbi12i 1532	Equality property for exclusive disjunction. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 21-Apr-2024.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃))
Theorem	xorbi12d 1533	Equality property for exclusive disjunction. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 ⊻ 𝜃) ↔ (𝜒 ⊻ 𝜏)))
Theorem	anxordi 1534	Conjunction distributes over exclusive-or. In intuitionistic logic this assertion is also true, even though xordi 1025 does not necessarily hold, in part because the usual definition of xor is subtly different in intuitionistic logic. (Contributed by David A. Wheeler, 7-Oct-2018.)
⊢ ((𝜑 ∧ (𝜓 ⊻ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒)))
Theorem	xorexmid 1535	Exclusive-or variant of the law of the excluded middle (exmid 901). This statement is ancient, going back to at least Stoic logic. This statement does not necessarily hold in intuitionistic logic. (Contributed by David A. Wheeler, 23-Feb-2019.)
⊢ (𝜑 ⊻ ¬ 𝜑)
1.2.14  Logical "nor"
Syntax	wnor 1536	Extend wff definition to include joint denial ("nor").
wff (𝜑 ⊽ 𝜓)
Definition	df-nor 1537	Define joint denial ("not-or" or "nor"). After we define the constant true ⊤ (df-tru 1551) and the constant false ⊥ (df-fal 1561), we will be able to prove these truth table values: ((⊤ ⊽ ⊤) ↔ ⊥) (trunortru 1597), ((⊤ ⊽ ⊥) ↔ ⊥) (trunorfal 1598), ((⊥ ⊽ ⊤) ↔ ⊥) (falnortru 1599), and ((⊥ ⊽ ⊥) ↔ ⊤) (falnorfal 1600). Contrast with ∧ (df-an 398), ∨ (df-or 855), → (wi 4), ⊼ (df-nan 1500), and ⊻ (df-xor 1520). (Contributed by Remi, 25-Oct-2023.)
⊢ ((𝜑 ⊽ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓))
Theorem	norcom 1538	The connector ⊽ is commutative. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 23-Apr-2024.)
⊢ ((𝜑 ⊽ 𝜓) ↔ (𝜓 ⊽ 𝜑))
Theorem	nornot 1539	¬ is expressible via ⊽. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
⊢ (¬ 𝜑 ↔ (𝜑 ⊽ 𝜑))
Theorem	noran 1540	∧ is expressible via ⊽. (Contributed by Remi, 26-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
⊢ ((𝜑 ∧ 𝜓) ↔ ((𝜑 ⊽ 𝜑) ⊽ (𝜓 ⊽ 𝜓)))
Theorem	noror 1541	∨ is expressible via ⊽. (Contributed by Remi, 26-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
⊢ ((𝜑 ∨ 𝜓) ↔ ((𝜑 ⊽ 𝜓) ⊽ (𝜑 ⊽ 𝜓)))
Theorem	norasslem1 1542	This lemma shows the equivalence of two expressions, used in norass 1545. (Contributed by Wolf Lammen, 18-Dec-2023.)
⊢ (((𝜑 ∨ 𝜓) → 𝜒) ↔ ((𝜑 ⊽ 𝜓) ∨ 𝜒))
Theorem	norasslem2 1543	This lemma specializes biimt 362 suitably for the proof of norass 1545. (Contributed by Wolf Lammen, 18-Dec-2023.)
⊢ (𝜑 → (𝜓 ↔ ((𝜑 ∨ 𝜒) → 𝜓)))
Theorem	norasslem3 1544	This lemma specializes biorf 943 suitably for the proof of norass 1545. (Contributed by Wolf Lammen, 18-Dec-2023.)
⊢ (¬ 𝜑 → ((𝜓 → 𝜒) ↔ ((𝜑 ∨ 𝜓) → 𝜒)))
Theorem	norass 1545	A characterization of when an expression involving joint denials associates. This is identical to the case when alternative denial is associative, see nanass 1518. Remark: Like alternative denial, joint denial is also commutative, see norcom 1538. (Contributed by RP, 29-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
⊢ ((𝜑 ↔ 𝜒) ↔ (((𝜑 ⊽ 𝜓) ⊽ 𝜒) ↔ (𝜑 ⊽ (𝜓 ⊽ 𝜒))))
1.2.15  True and false constants
1.2.15.1  Universal quantifier for use by df-tru Even though it is not ordinarily part of propositional calculus, the universal quantifier ∀ is introduced here so that the soundness of Definition df-tru 1551 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in Definition df-ex 1788 in the predicate calculus section below. For those who want propositional calculus to be self-contained, i.e., to use wff variables only, the alternate Definition dftru2 1553 may be adopted and this subsection moved down to the start of the subsection with wex 1787 below. However, the use of dftru2 1553 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.
Syntax	wal 1546	Extend wff definition to include the universal quantifier ("for all"). ∀𝑥𝜑 is read "𝜑 (phi) is true for all 𝑥". Typically, in its final application 𝜑 would be replaced with a wff containing a (free) occurrence of the variable 𝑥, for example 𝑥 = 𝑦. In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of 𝑥. When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same.
wff ∀𝑥𝜑
1.2.15.2  Equality predicate for use by df-tru Even though it is not ordinarily part of propositional calculus, the equality predicate = is introduced here so that the soundness of definition df-tru 1551 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in Theorem weq 1970 in the predicate calculus section below. For those who want propositional calculus to be self-contained, i.e., to use wff variables only, the alternate definition dftru2 1553 may be adopted and this subsection moved down to just above weq 1970 below. However, the use of dftru2 1553 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.
Syntax	cv 1547	This syntax construction states that a variable 𝑥, which has been declared to be a setvar variable by $f statement vx, is also a class expression. This can be justified informally as follows. We know that the class builder {𝑦 ∣ 𝑦 ∈ 𝑥} is a class by cab 2719. Since (when 𝑦 is distinct from 𝑥) we have 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} by cvjust 2735, we can argue that the syntax "class 𝑥 " can be viewed as an abbreviation for "class {𝑦 ∣ 𝑦 ∈ 𝑥}". See the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class". While it is tempting and perhaps occasionally useful to view cv 1547 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1547 is intrinsically no different from any other class-building syntax such as cab 2719, cun 3882, or c0 4263. For a general discussion of the theory of classes and the role of cv 1547, see mmset.html#class 1547. (The description above applies to set theory, not predicate calculus. The purpose of introducing class 𝑥 here, and not in set theory where it belongs, is to allow to express, i.e., "prove", the weq 1970 of predicate calculus from the wceq 1548 of set theory, so that we do not overload the = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.)
class 𝑥
Syntax	wceq 1548	Extend wff definition to include class equality. For a general discussion of the theory of classes, see mmset.html#class. (The purpose of introducing wff 𝐴 = 𝐵 here, and not in set theory where it belongs, is to allow to express, i.e., "prove", the weq 1970 of predicate calculus in terms of the wceq 1548 of set theory, so that we do not "overload" the = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. For example, some parsers - although not the Metamath program - stumble on the fact that the = in 𝑥 = 𝑦 could be the = of either weq 1970 or wceq 1548, although mathematically it makes no difference. The class variables 𝐴 and 𝐵 are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 2733 for more information on the set theory usage of wceq 1548.)
wff 𝐴 = 𝐵
1.2.15.3  The true constant
Syntax	wtru 1549	The constant ⊤ is a wff.
wff ⊤
Theorem	trujust 1550	Soundness justification theorem for df-tru 1551. Instance of monothetic 268. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.)
⊢ ((∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥) ↔ (∀𝑦 𝑦 = 𝑦 → ∀𝑦 𝑦 = 𝑦))
Definition	df-tru 1551	Definition of the truth value "true", or "verum", denoted by ⊤. In this definition, an instance of id 22 is used as the definiens, although any tautology, such as an axiom, can be used in its place. This particular instance of id 22 was chosen so this definition can be checked by the same algorithm that is used for predicate calculus. This definition should be referenced directly only by tru 1552, and other proofs should use tru 1552 instead of this definition, since there are many alternate ways to define ⊤. (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by NM, 11-Jul-2019.) Use tru 1552 instead. (New usage is discouraged.)
⊢ (⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
Theorem	tru 1552	The truth value ⊤ is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
⊢ ⊤
Theorem	dftru2 1553	An alternate definition of "true" (see comment of df-tru 1551). The associated justification theorem is monothetic 268. (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) Use tru 1552 instead. (New usage is discouraged.)
⊢ (⊤ ↔ (𝜑 → 𝜑))
Theorem	trut 1554	A proposition is equivalent to it being implied by ⊤. Closed form of mptru 1555. Dual of dfnot 1567. It is to tbtru 1556 what a1bi 364 is to tbt 371. (Contributed by BJ, 26-Oct-2019.)
⊢ (𝜑 ↔ (⊤ → 𝜑))
Theorem	mptru 1555	Eliminate ⊤ as an antecedent. A proposition implied by ⊤ is true. This is modus ponens ax-mp 5 when the minor hypothesis is ⊤ (which holds by tru 1552). (Contributed by Mario Carneiro, 13-Mar-2014.)
⊢ (⊤ → 𝜑)    ⇒   ⊢ 𝜑
Theorem	tbtru 1556	A proposition is equivalent to itself being equivalent to ⊤. (Contributed by Anthony Hart, 14-Aug-2011.)
⊢ (𝜑 ↔ (𝜑 ↔ ⊤))
Theorem	bitru 1557	A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
⊢ 𝜑    ⇒   ⊢ (𝜑 ↔ ⊤)
Theorem	trud 1558	Anything implies ⊤. Dual statement of falim 1565. Deduction form of tru 1552. Note on naming: in 2022, the theorem now known as mptru 1555 was renamed from trud so if you are reading documentation written before that time, references to trud refer to what is now mptru 1555. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
⊢ (𝜑 → ⊤)
Theorem	truan 1559	True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
⊢ ((⊤ ∧ 𝜑) ↔ 𝜑)
1.2.15.4  The false constant
Syntax	wfal 1560	The constant ⊥ is a wff.
wff ⊥
Definition	df-fal 1561	Definition of the truth value "false", or "falsum", denoted by ⊥. See also df-tru 1551. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ (⊥ ↔ ¬ ⊤)
Theorem	fal 1562	The truth value ⊥ is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
⊢ ¬ ⊥
Theorem	nbfal 1563	The negation of a proposition is equivalent to itself being equivalent to ⊥. (Contributed by Anthony Hart, 14-Aug-2011.)
⊢ (¬ 𝜑 ↔ (𝜑 ↔ ⊥))
Theorem	bifal 1564	A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
⊢ ¬ 𝜑    ⇒   ⊢ (𝜑 ↔ ⊥)
Theorem	falim 1565	The truth value ⊥ implies anything. Also called the "principle of explosion", or "ex falso [sequitur]] quodlibet" (Latin for "from falsehood, anything [follows]]"). Dual statement of trud 1558. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
⊢ (⊥ → 𝜑)
Theorem	falimd 1566	The truth value ⊥ implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ ((𝜑 ∧ ⊥) → 𝜓)
Theorem	dfnot 1567	Given falsum ⊥, we can define the negation of a wff 𝜑 as the statement that ⊥ follows from assuming 𝜑. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
⊢ (¬ 𝜑 ↔ (𝜑 → ⊥))
Theorem	inegd 1568	Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ ((𝜑 ∧ 𝜓) → ⊥)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	efald 1569	Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ ((𝜑 ∧ ¬ 𝜓) → ⊥)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.21fal 1570	If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ⊥)
1.2.16  Truth tables Some sources define logical connectives by their truth tables. These are tables that give the truth value of the composed expression for all possible combinations of the truth values of their arguments. In this section, we show that our definitions and axioms produce equivalent results for all the logical connectives we have introduced (either axiomatically or by a definition): implication wi 4, negation wn 3, biconditional df-bi 209, conjunction df-an 398, disjunction df-or 855, alternative denial df-nan 1500, exclusive disjunction df-xor 1520.
1.2.16.1  Implication
Theorem	truimtru 1571	A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using trud 1558 instead of id 22 but the principle of identity id 22 is more basic, and the present proof indicates that the result still holds in relevance logic. (Proof modification is discouraged.)
⊢ ((⊤ → ⊤) ↔ ⊤)
Theorem	truimfal 1572	A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊤ → ⊥) ↔ ⊥)
Theorem	falimtru 1573	A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using falim 1565 instead of trud 1558 but the present proof using trud 1558 emphasizes that the result does not require the principle of explosion. (Proof modification is discouraged.)
⊢ ((⊥ → ⊤) ↔ ⊤)
Theorem	falimfal 1574	A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using falim 1565 instead of id 22 but the present proof using id 22 emphasizes that the result does not require the principle of explosion. (Proof modification is discouraged.)
⊢ ((⊥ → ⊥) ↔ ⊤)
1.2.16.2  Negation
Theorem	nottru 1575	A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ (¬ ⊤ ↔ ⊥)
Theorem	notfal 1576	A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ (¬ ⊥ ↔ ⊤)
1.2.16.3  Equivalence
Theorem	trubitru 1577	A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊤ ↔ ⊤) ↔ ⊤)
Theorem	falbitru 1578	A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
⊢ ((⊥ ↔ ⊤) ↔ ⊥)
Theorem	trubifal 1579	A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
⊢ ((⊤ ↔ ⊥) ↔ ⊥)
Theorem	falbifal 1580	A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊥ ↔ ⊥) ↔ ⊤)
1.2.16.4  Conjunction
Theorem	truantru 1581	A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊤ ∧ ⊤) ↔ ⊤)
Theorem	truanfal 1582	A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊤ ∧ ⊥) ↔ ⊥)
Theorem	falantru 1583	A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊥ ∧ ⊤) ↔ ⊥)
Theorem	falanfal 1584	A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊥ ∧ ⊥) ↔ ⊥)
1.2.16.5  Disjunction
Theorem	truortru 1585	A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊤ ∨ ⊤) ↔ ⊤)
Theorem	truorfal 1586	A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊤ ∨ ⊥) ↔ ⊤)
Theorem	falortru 1587	A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊥ ∨ ⊤) ↔ ⊤)
Theorem	falorfal 1588	A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊥ ∨ ⊥) ↔ ⊥)
1.2.16.6  Alternative denial
Theorem	trunantru 1589	A ⊼ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊤ ⊼ ⊤) ↔ ⊥)
Theorem	trunanfal 1590	A ⊼ identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
⊢ ((⊤ ⊼ ⊥) ↔ ⊤)
Theorem	falnantru 1591	A ⊼ identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊥ ⊼ ⊤) ↔ ⊤)
Theorem	falnanfal 1592	A ⊼ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊥ ⊼ ⊥) ↔ ⊤)
1.2.16.7  Exclusive disjunction
Theorem	truxortru 1593	A ⊻ identity. (Contributed by David A. Wheeler, 8-May-2015.)
⊢ ((⊤ ⊻ ⊤) ↔ ⊥)
Theorem	truxorfal 1594	A ⊻ identity. (Contributed by David A. Wheeler, 8-May-2015.)
⊢ ((⊤ ⊻ ⊥) ↔ ⊤)
Theorem	falxortru 1595	A ⊻ identity. (Contributed by David A. Wheeler, 9-May-2015.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
⊢ ((⊥ ⊻ ⊤) ↔ ⊤)
Theorem	falxorfal 1596	A ⊻ identity. (Contributed by David A. Wheeler, 9-May-2015.)
⊢ ((⊥ ⊻ ⊥) ↔ ⊥)
1.2.16.8  Joint denial
Theorem	trunortru 1597	A ⊽ identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 7-Dec-2023.)
⊢ ((⊤ ⊽ ⊤) ↔ ⊥)
Theorem	trunorfal 1598	A ⊽ identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
⊢ ((⊤ ⊽ ⊥) ↔ ⊥)
Theorem	falnortru 1599	A ⊽ identity. (Contributed by Remi, 25-Oct-2023.)
⊢ ((⊥ ⊽ ⊤) ↔ ⊥)
Theorem	falnorfal 1600	A ⊽ identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
⊢ ((⊥ ⊽ ⊥) ↔ ⊤)