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	nanbi2i 1501	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 1502	Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜃))
Theorem	nanbi1d 1503	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 1504	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 1505	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 1506	A characterization of when an expression involving alternative denials associates. Remark: alternative denial is commutative, see nancom 1492. (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 1507	Extend wff definition to include exclusive disjunction ("xor").
wff (𝜑 ⊻ 𝜓)
Definition	df-xor 1508	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 1539) and the constant false ⊥ (df-fal 1549), we will be able to prove these truth table values: ((⊤ ⊻ ⊤) ↔ ⊥) (truxortru 1581), ((⊤ ⊻ ⊥) ↔ ⊤) (truxorfal 1582), ((⊥ ⊻ ⊤) ↔ ⊤) (falxortru 1583), and ((⊥ ⊻ ⊥) ↔ ⊥) (falxorfal 1584). Contrast with ∧ (df-an 396), ∨ (df-or 848), → (wi 4), and ⊼ (df-nan 1488). (Contributed by FL, 22-Nov-2010.)
⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
Theorem	xnor 1509	Two ways to write XNOR (exclusive not-or). (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓))
Theorem	xorcom 1510	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 1511	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 1512	This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.)
⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓)))
Theorem	xor2 1513	Two ways to express "exclusive or". (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
Theorem	xoror 1514	Exclusive disjunction implies disjunction ("XOR implies OR"). (Contributed by BJ, 19-Apr-2019.)
⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ∨ 𝜓))
Theorem	xornan 1515	Exclusive disjunction implies alternative denial ("XOR implies NAND"). (Contributed by BJ, 19-Apr-2019.)
⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ∧ 𝜓))
Theorem	xornan2 1516	XOR implies NAND (written with the ⊼ connector). (Contributed by BJ, 19-Apr-2019.)
⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ⊼ 𝜓))
Theorem	xorneg2 1517	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 1518	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 1519	The connector ⊻ is unchanged under negation of both arguments. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ ((¬ 𝜑 ⊻ ¬ 𝜓) ↔ (𝜑 ⊻ 𝜓))
Theorem	xorbi12i 1520	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 1521	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 1522	Conjunction distributes over exclusive-or. In intuitionistic logic this assertion is also true, even though xordi 1018 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 1523	Exclusive-or variant of the law of the excluded middle (exmid 894). 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 1524	Extend wff definition to include joint denial ("nor").
wff (𝜑 ⊽ 𝜓)
Definition	df-nor 1525	Define joint denial ("not-or" or "nor"). After we define the constant true ⊤ (df-tru 1539) and the constant false ⊥ (df-fal 1549), we will be able to prove these truth table values: ((⊤ ⊽ ⊤) ↔ ⊥) (trunortru 1585), ((⊤ ⊽ ⊥) ↔ ⊥) (trunorfal 1586), ((⊥ ⊽ ⊤) ↔ ⊥) (falnortru 1587), and ((⊥ ⊽ ⊥) ↔ ⊤) (falnorfal 1588). Contrast with ∧ (df-an 396), ∨ (df-or 848), → (wi 4), ⊼ (df-nan 1488), and ⊻ (df-xor 1508). (Contributed by Remi, 25-Oct-2023.)
⊢ ((𝜑 ⊽ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓))
Theorem	norcom 1526	The connector ⊽ is commutative. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 23-Apr-2024.)
⊢ ((𝜑 ⊽ 𝜓) ↔ (𝜓 ⊽ 𝜑))
Theorem	nornot 1527	¬ is expressible via ⊽. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
⊢ (¬ 𝜑 ↔ (𝜑 ⊽ 𝜑))
Theorem	noran 1528	∧ 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 1529	∨ 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 1530	This lemma shows the equivalence of two expressions, used in norass 1533. (Contributed by Wolf Lammen, 18-Dec-2023.)
⊢ (((𝜑 ∨ 𝜓) → 𝜒) ↔ ((𝜑 ⊽ 𝜓) ∨ 𝜒))
Theorem	norasslem2 1531	This lemma specializes biimt 360 suitably for the proof of norass 1533. (Contributed by Wolf Lammen, 18-Dec-2023.)
⊢ (𝜑 → (𝜓 ↔ ((𝜑 ∨ 𝜒) → 𝜓)))
Theorem	norasslem3 1532	This lemma specializes biorf 936 suitably for the proof of norass 1533. (Contributed by Wolf Lammen, 18-Dec-2023.)
⊢ (¬ 𝜑 → ((𝜓 → 𝜒) ↔ ((𝜑 ∨ 𝜓) → 𝜒)))
Theorem	norass 1533	A characterization of when an expression involving joint denials associates. This is identical to the case when alternative denial is associative, see nanass 1506. Remark: Like alternative denial, joint denial is also commutative, see norcom 1526. (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 1539 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in Definition df-ex 1776 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 1541 may be adopted and this subsection moved down to the start of the subsection with wex 1775 below. However, the use of dftru2 1541 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.
Syntax	wal 1534	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 1539 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in Theorem weq 1959 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 1541 may be adopted and this subsection moved down to just above weq 1959 below. However, the use of dftru2 1541 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.
Syntax	cv 1535	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 2711. Since (when 𝑦 is distinct from 𝑥) we have 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} by cvjust 2728, 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 1535 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1535 is intrinsically no different from any other class-building syntax such as cab 2711, cun 3960, or c0 4338. For a general discussion of the theory of classes and the role of cv 1535, see mmset.html#class 1535. (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 1959 of predicate calculus from the wceq 1536 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 1536	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 1959 of predicate calculus in terms of the wceq 1536 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 1959 or wceq 1536, 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 2726 for more information on the set theory usage of wceq 1536.)
wff 𝐴 = 𝐵
1.2.15.3  The true constant
Syntax	wtru 1537	The constant ⊤ is a wff.
wff ⊤
Theorem	trujust 1538	Soundness justification theorem for df-tru 1539. Instance of monothetic 266. (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 1539	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 1540, and other proofs should use tru 1540 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 1540 instead. (New usage is discouraged.)
⊢ (⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
Theorem	tru 1540	The truth value ⊤ is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
⊢ ⊤
Theorem	dftru2 1541	An alternate definition of "true" (see comment of df-tru 1539). The associated justification theorem is monothetic 266. (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) Use tru 1540 instead. (New usage is discouraged.)
⊢ (⊤ ↔ (𝜑 → 𝜑))
Theorem	trut 1542	A proposition is equivalent to it being implied by ⊤. Closed form of mptru 1543. Dual of dfnot 1555. It is to tbtru 1544 what a1bi 362 is to tbt 369. (Contributed by BJ, 26-Oct-2019.)
⊢ (𝜑 ↔ (⊤ → 𝜑))
Theorem	mptru 1543	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 1540). (Contributed by Mario Carneiro, 13-Mar-2014.)
⊢ (⊤ → 𝜑)    ⇒   ⊢ 𝜑
Theorem	tbtru 1544	A proposition is equivalent to itself being equivalent to ⊤. (Contributed by Anthony Hart, 14-Aug-2011.)
⊢ (𝜑 ↔ (𝜑 ↔ ⊤))
Theorem	bitru 1545	A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
⊢ 𝜑    ⇒   ⊢ (𝜑 ↔ ⊤)
Theorem	trud 1546	Anything implies ⊤. Dual statement of falim 1553. Deduction form of tru 1540. Note on naming: in 2022, the theorem now known as mptru 1543 was renamed from trud so if you are reading documentation written before that time, references to trud refer to what is now mptru 1543. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
⊢ (𝜑 → ⊤)
Theorem	truan 1547	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 1548	The constant ⊥ is a wff.
wff ⊥
Definition	df-fal 1549	Definition of the truth value "false", or "falsum", denoted by ⊥. See also df-tru 1539. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ (⊥ ↔ ¬ ⊤)
Theorem	fal 1550	The truth value ⊥ is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
⊢ ¬ ⊥
Theorem	nbfal 1551	The negation of a proposition is equivalent to itself being equivalent to ⊥. (Contributed by Anthony Hart, 14-Aug-2011.)
⊢ (¬ 𝜑 ↔ (𝜑 ↔ ⊥))
Theorem	bifal 1552	A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
⊢ ¬ 𝜑    ⇒   ⊢ (𝜑 ↔ ⊥)
Theorem	falim 1553	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 1546. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
⊢ (⊥ → 𝜑)
Theorem	falimd 1554	The truth value ⊥ implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ ((𝜑 ∧ ⊥) → 𝜓)
Theorem	dfnot 1555	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 1556	Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ ((𝜑 ∧ 𝜓) → ⊥)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	efald 1557	Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ ((𝜑 ∧ ¬ 𝜓) → ⊥)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.21fal 1558	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 207, conjunction df-an 396, disjunction df-or 848, alternative denial df-nan 1488, exclusive disjunction df-xor 1508.
1.2.16.1  Implication
Theorem	truimtru 1559	A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using trud 1546 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 1560	A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊤ → ⊥) ↔ ⊥)
Theorem	falimtru 1561	A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using falim 1553 instead of trud 1546 but the present proof using trud 1546 emphasizes that the result does not require the principle of explosion. (Proof modification is discouraged.)
⊢ ((⊥ → ⊤) ↔ ⊤)
Theorem	falimfal 1562	A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using falim 1553 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 1563	A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ (¬ ⊤ ↔ ⊥)
Theorem	notfal 1564	A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ (¬ ⊥ ↔ ⊤)
1.2.16.3  Equivalence
Theorem	trubitru 1565	A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊤ ↔ ⊤) ↔ ⊤)
Theorem	falbitru 1566	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 1567	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 1568	A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊥ ↔ ⊥) ↔ ⊤)
1.2.16.4  Conjunction
Theorem	truantru 1569	A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊤ ∧ ⊤) ↔ ⊤)
Theorem	truanfal 1570	A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊤ ∧ ⊥) ↔ ⊥)
Theorem	falantru 1571	A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊥ ∧ ⊤) ↔ ⊥)
Theorem	falanfal 1572	A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊥ ∧ ⊥) ↔ ⊥)
1.2.16.5  Disjunction
Theorem	truortru 1573	A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊤ ∨ ⊤) ↔ ⊤)
Theorem	truorfal 1574	A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊤ ∨ ⊥) ↔ ⊤)
Theorem	falortru 1575	A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊥ ∨ ⊤) ↔ ⊤)
Theorem	falorfal 1576	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 1577	A ⊼ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊤ ⊼ ⊤) ↔ ⊥)
Theorem	trunanfal 1578	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 1579	A ⊼ identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊥ ⊼ ⊤) ↔ ⊤)
Theorem	falnanfal 1580	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 1581	A ⊻ identity. (Contributed by David A. Wheeler, 8-May-2015.)
⊢ ((⊤ ⊻ ⊤) ↔ ⊥)
Theorem	truxorfal 1582	A ⊻ identity. (Contributed by David A. Wheeler, 8-May-2015.)
⊢ ((⊤ ⊻ ⊥) ↔ ⊤)
Theorem	falxortru 1583	A ⊻ identity. (Contributed by David A. Wheeler, 9-May-2015.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
⊢ ((⊥ ⊻ ⊤) ↔ ⊤)
Theorem	falxorfal 1584	A ⊻ identity. (Contributed by David A. Wheeler, 9-May-2015.)
⊢ ((⊥ ⊻ ⊥) ↔ ⊥)
1.2.16.8  Joint denial
Theorem	trunortru 1585	A ⊽ identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 7-Dec-2023.)
⊢ ((⊤ ⊽ ⊤) ↔ ⊥)
Theorem	trunorfal 1586	A ⊽ identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
⊢ ((⊤ ⊽ ⊥) ↔ ⊥)
Theorem	falnortru 1587	A ⊽ identity. (Contributed by Remi, 25-Oct-2023.)
⊢ ((⊥ ⊽ ⊤) ↔ ⊥)
Theorem	falnorfal 1588	A ⊽ identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
⊢ ((⊥ ⊽ ⊥) ↔ ⊤)
1.2.17  Half adder and full adder in propositional calculus Propositional calculus deals with truth values, which can be interpreted as bits. Using this, we can define the half adder and the full adder in pure propositional calculus, and show their basic properties. The half adder adds two 1-bit numbers. Its two outputs are the "sum" S and the "carry" C. The real sum is then given by 2C+S. The sum and carry correspond respectively to the logical exclusive disjunction (df-xor 1508) and the logical conjunction (df-an 396). The full adder takes into account an "input carry", so it has three inputs and again two outputs, corresponding to the "sum" (df-had 1590) and "updated carry" (df-cad 1603). Here is a short description. We code the bit 0 by ⊥ and 1 by ⊤. Even though hadd and cadd are invariant under permutation of their arguments, assume for the sake of concreteness that 𝜑 (resp. 𝜓) is the i^th bit of the first (resp. second) number to add (with the convention that the i^th bit is the multiple of 2^i in the base-2 representation), and that 𝜒 is the i^th carry (with the convention that the 0^th carry is 0). Then, hadd(𝜑, 𝜓, 𝜒) gives the i^th bit of the sum, and cadd(𝜑, 𝜓, 𝜒) gives the (i+1)^th carry. Then, addition is performed by iteration from i = 0 to i = 1 + (max of the number of digits of the two summands) by "updating" the carry.
1.2.17.1  Full adder: sum
Syntax	whad 1589	Syntax for the "sum" output of the full adder. (Contributed by Mario Carneiro, 4-Sep-2016.)
wff hadd(𝜑, 𝜓, 𝜒)
Definition	df-had 1590	Definition of the "sum" output of the full adder (triple exclusive disjunction, or XOR3, or testing whether an odd number of parameters are true). (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ⊻ 𝜓) ⊻ 𝜒))
Theorem	hadbi123d 1591	Equality theorem for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    &   ⊢ (𝜑 → (𝜂 ↔ 𝜁))    ⇒   ⊢ (𝜑 → (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁)))
Theorem	hadbi123i 1592	Equality theorem for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    &   ⊢ (𝜏 ↔ 𝜂)    ⇒   ⊢ (hadd(𝜑, 𝜒, 𝜏) ↔ hadd(𝜓, 𝜃, 𝜂))
Theorem	hadass 1593	Associative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓 ⊻ 𝜒)))
Theorem	hadbi 1594	The adder sum is the same as the triple biconditional. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ 𝜒))
Theorem	hadcoma 1595	Commutative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜑, 𝜒))
Theorem	hadcomb 1596	Commutative law for the adders sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜑, 𝜒, 𝜓))
Theorem	hadrot 1597	Rotation law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜒, 𝜑))
Theorem	hadnot 1598	The adder sum distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)
⊢ (¬ hadd(𝜑, 𝜓, 𝜒) ↔ hadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒))
Theorem	had1 1599	If the first input is true, then the adder sum is equivalent to the biconditionality of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)
⊢ (𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ↔ 𝜒)))
Theorem	had0 1600	If the first input is false, then the adder sum is equivalent to the exclusive disjunction of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jul-2020.)
⊢ (¬ 𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ⊻ 𝜒)))