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Theorem List for Metamath Proof Explorer - 1501-1600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnanbi12i 1501 Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))
 
Theoremnanbi1d 1502 Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))
 
Theoremnanbi2d 1503 Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))
 
Theoremnanbi12d 1504 Join two logical equivalences with anti-conjunction. (Contributed by Scott Fenton, 2-Jan-2018.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theoremnanass 1505 A characterization of when an expression involving alternative denials associates. Remark: alternative denial is commutative, see nancom 1491. (Contributed by Richard Penner, 29-Feb-2020.) (Proof shortened by Wolf Lammen, 23-Oct-2022.)
((𝜑𝜒) ↔ (((𝜑𝜓) ⊼ 𝜒) ↔ (𝜑 ⊼ (𝜓𝜒))))
 
1.2.13  Logical "xor"
 
Syntaxwxo 1506 Extend wff definition to include exclusive disjunction ("xor").
wff (𝜑𝜓)
 
Definitiondf-xor 1507 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 1545) and the constant false (df-fal 1555), we will be able to prove these truth table values: ((⊤ ⊻ ⊤) ↔ ⊥) (truxortru 1587), ((⊤ ⊻ ⊥) ↔ ⊤) (truxorfal 1588), ((⊥ ⊻ ⊤) ↔ ⊤) (falxortru 1589), and ((⊥ ⊻ ⊥) ↔ ⊥) (falxorfal 1590). Contrast with (df-an 397), (df-or 845), (wi 4), and (df-nan 1487). (Contributed by FL, 22-Nov-2010.)
((𝜑𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremxnor 1508 Two ways to write XNOR (exclusive not-or). (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremxorcom 1509 The connector is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 21-Apr-2024.)
((𝜑𝜓) ↔ (𝜓𝜑))
 
TheoremxorcomOLD 1510 Obsolete version of xorcom 1509 as of 21-Apr-2024. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ↔ (𝜓𝜑))
 
Theoremxorass 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.)
(((𝜑𝜓) ⊻ 𝜒) ↔ (𝜑 ⊻ (𝜓𝜒)))
 
Theoremexcxor 1512 This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.)
((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
 
Theoremxor2 1513 Two ways to express "exclusive or". (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
 
Theoremxoror 1514 Exclusive disjunction implies disjunction ("XOR implies OR"). (Contributed by BJ, 19-Apr-2019.)
((𝜑𝜓) → (𝜑𝜓))
 
Theoremxornan 1515 Exclusive disjunction implies alternative denial ("XOR implies NAND"). (Contributed by BJ, 19-Apr-2019.)
((𝜑𝜓) → ¬ (𝜑𝜓))
 
Theoremxornan2 1516 XOR implies NAND (written with the connector). (Contributed by BJ, 19-Apr-2019.)
((𝜑𝜓) → (𝜑𝜓))
 
Theoremxorneg2 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) ↔ ¬ (𝜑𝜓))
 
Theoremxorneg1 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.)
((¬ 𝜑𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremxorneg 1519 The connector is unchanged under negation of both arguments. (Contributed by Mario Carneiro, 4-Sep-2016.)
((¬ 𝜑 ⊻ ¬ 𝜓) ↔ (𝜑𝜓))
 
Theoremxorbi12i 1520 Equality property for exclusive disjunction. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 21-Apr-2024.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))
 
Theoremxorbi12iOLD 1521 Obsolete version of xorbi12i 1520 as of 21-Apr-2024. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))
 
Theoremxorbi12d 1522 Equality property for exclusive disjunction. (Contributed by Mario Carneiro, 4-Sep-2016.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theoremanxordi 1523 Conjunction distributes over exclusive-or. In intuitionistic logic this assertion is also true, even though xordi 1014 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 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒)))
 
Theoremxorexmid 1524 Exclusive-or variant of the law of the excluded middle (exmid 892). 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"
 
Syntaxwnor 1525 Extend wff definition to include joint denial ("nor").
wff (𝜑 𝜓)
 
Definitiondf-nor 1526 Define joint denial ("not-or" or "nor"). After we define the constant true (df-tru 1545) and the constant false (df-fal 1555), we will be able to prove these truth table values: ((⊤ ⊤) ↔ ⊥) (trunortru 1591), ((⊤ ⊥) ↔ ⊥) (trunorfal 1593), ((⊥ ⊤) ↔ ⊥) (falnortru 1595), and ((⊥ ⊥) ↔ ⊤) (falnorfal 1596). Contrast with (df-an 397), (df-or 845), (wi 4), (df-nan 1487), and (df-xor 1507). (Contributed by Remi, 25-Oct-2023.)
((𝜑 𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremnorcom 1527 The connector is commutative. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 23-Apr-2024.)
((𝜑 𝜓) ↔ (𝜓 𝜑))
 
TheoremnorcomOLD 1528 Obsolete version of norcom 1527 as of 23-Apr-2024. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 𝜓) ↔ (𝜓 𝜑))
 
Theoremnornot 1529 ¬ is expressible via . (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
𝜑 ↔ (𝜑 𝜑))
 
TheoremnornotOLD 1530 Obsolete version of nornot 1529 as of 8-Dec-2023. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑 ↔ (𝜑 𝜑))
 
Theoremnoran 1531 is expressible via . (Contributed by Remi, 26-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
((𝜑𝜓) ↔ ((𝜑 𝜑) (𝜓 𝜓)))
 
TheoremnoranOLD 1532 Obsolete version of noran 1531 as of 8-Dec-2023. (Contributed by Remi, 26-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ↔ ((𝜑 𝜑) (𝜓 𝜓)))
 
Theoremnoror 1533 is expressible via . (Contributed by Remi, 26-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
((𝜑𝜓) ↔ ((𝜑 𝜓) (𝜑 𝜓)))
 
TheoremnororOLD 1534 Obsolete version of noror 1533 as of 8-Dec-2023. (Contributed by Remi, 26-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ↔ ((𝜑 𝜓) (𝜑 𝜓)))
 
Theoremnorasslem1 1535 This lemma shows the equivalence of two expressions, used in norass 1538. (Contributed by Wolf Lammen, 18-Dec-2023.)
(((𝜑𝜓) → 𝜒) ↔ ((𝜑 𝜓) ∨ 𝜒))
 
Theoremnorasslem2 1536 This lemma specializes biimt 361 suitably for the proof of norass 1538. (Contributed by Wolf Lammen, 18-Dec-2023.)
(𝜑 → (𝜓 ↔ ((𝜑𝜒) → 𝜓)))
 
Theoremnorasslem3 1537 This lemma specializes biorf 934 suitably for the proof of norass 1538. (Contributed by Wolf Lammen, 18-Dec-2023.)
𝜑 → ((𝜓𝜒) ↔ ((𝜑𝜓) → 𝜒)))
 
Theoremnorass 1538 A characterization of when an expression involving joint denials associates. This is identical to the case when alternative denial is associative, see nanass 1505. Remark: Like alternative denial, joint denial is also commutative, see norcom 1527. (Contributed by RP, 29-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
((𝜑𝜒) ↔ (((𝜑 𝜓) 𝜒) ↔ (𝜑 (𝜓 𝜒))))
 
TheoremnorassOLD 1539 Obsolete version of norass 1538 as of 17-Dec-2023. (Contributed by RP, 29-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜒) ↔ (((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 1545 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in Definition df-ex 1787 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 1547 may be adopted and this subsection moved down to the start of the subsection with wex 1786 below. However, the use of dftru2 1547 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

 
Syntaxwal 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
 
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 1545 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 1547 may be adopted and this subsection moved down to just above weq 1970 below. However, the use of dftru2 1547 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

 
Syntaxcv 1541 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 {𝑦𝑦𝑥} is a class by cab 2717. Since (when 𝑦 is distinct from 𝑥) we have 𝑥 = {𝑦𝑦𝑥} by cvjust 2734, we can argue that the syntax "class 𝑥 " can be viewed as an abbreviation for "class {𝑦𝑦𝑥}". 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 1541 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1541 is intrinsically no different from any other class-building syntax such as cab 2717, cun 3890, or c0 4262.

For a general discussion of the theory of classes and the role of cv 1541, see mmset.html#class 1541.

(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 us to express, i.e., "prove", the weq 1970 of predicate calculus from the wceq 1542 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
 
Syntaxwceq 1542 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 us to express, i.e., "prove", the weq 1970 of predicate calculus in terms of the wceq 1542 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 1542, 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 2732 for more information on the set theory usage of wceq 1542.)

wff 𝐴 = 𝐵
 
1.2.15.3  The true constant
 
Syntaxwtru 1543 The constant is a wff.
wff
 
Theoremtrujust 1544 Soundness justification theorem for df-tru 1545. Instance of monothetic 265. (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))
 
Definitiondf-tru 1545 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 1546, and other proofs should use tru 1546 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 1546 instead. (New usage is discouraged.)
(⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
 
Theoremtru 1546 The truth value is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
 
Theoremdftru2 1547 An alternate definition of "true" (see comment of df-tru 1545). The associated justification theorem is monothetic 265. (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) Use tru 1546 instead. (New usage is discouraged.)
(⊤ ↔ (𝜑𝜑))
 
Theoremtrut 1548 A proposition is equivalent to it being implied by . Closed form of mptru 1549. Dual of dfnot 1561. It is to tbtru 1550 what a1bi 363 is to tbt 370. (Contributed by BJ, 26-Oct-2019.)
(𝜑 ↔ (⊤ → 𝜑))
 
Theoremmptru 1549 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 1546). (Contributed by Mario Carneiro, 13-Mar-2014.)
(⊤ → 𝜑)       𝜑
 
Theoremtbtru 1550 A proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.)
(𝜑 ↔ (𝜑 ↔ ⊤))
 
Theorembitru 1551 A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
𝜑       (𝜑 ↔ ⊤)
 
Theoremtrud 1552 Anything implies . Dual statement of falim 1559. Deduction form of tru 1546. Note on naming: in 2022, the theorem now known as mptru 1549 was renamed from trud so if you are reading documentation written before that time, references to trud refer to what is now mptru 1549. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
(𝜑 → ⊤)
 
Theoremtruan 1553 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
 
Syntaxwfal 1554 The constant is a wff.
wff
 
Definitiondf-fal 1555 Definition of the truth value "false", or "falsum", denoted by . See also df-tru 1545. (Contributed by Anthony Hart, 22-Oct-2010.)
(⊥ ↔ ¬ ⊤)
 
Theoremfal 1556 The truth value is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
¬ ⊥
 
Theoremnbfal 1557 The negation of a proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.)
𝜑 ↔ (𝜑 ↔ ⊥))
 
Theorembifal 1558 A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
¬ 𝜑       (𝜑 ↔ ⊥)
 
Theoremfalim 1559 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 1552. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
(⊥ → 𝜑)
 
Theoremfalimd 1560 The truth value implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑 ∧ ⊥) → 𝜓)
 
Theoremdfnot 1561 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 → ⊥))
 
Theoreminegd 1562 Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑𝜓) → ⊥)       (𝜑 → ¬ 𝜓)
 
Theoremefald 1563 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑 ∧ ¬ 𝜓) → ⊥)       (𝜑𝜓)
 
Theorempm2.21fal 1564 If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑𝜓)    &   (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 206, conjunction df-an 397, disjunction df-or 845, alternative denial df-nan 1487, exclusive disjunction df-xor 1507.

 
1.2.16.1  Implication
 
Theoremtruimtru 1565 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using trud 1552 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.)
((⊤ → ⊤) ↔ ⊤)
 
Theoremtruimfal 1566 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ → ⊥) ↔ ⊥)
 
Theoremfalimtru 1567 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using falim 1559 instead of trud 1552 but the present proof using trud 1552 emphasizes that the result does not require the principle of explosion. (Proof modification is discouraged.)
((⊥ → ⊤) ↔ ⊤)
 
Theoremfalimfal 1568 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using falim 1559 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
 
Theoremnottru 1569 A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(¬ ⊤ ↔ ⊥)
 
Theoremnotfal 1570 A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(¬ ⊥ ↔ ⊤)
 
1.2.16.3  Equivalence
 
Theoremtrubitru 1571 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ↔ ⊤) ↔ ⊤)
 
Theoremfalbitru 1572 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.)
((⊥ ↔ ⊤) ↔ ⊥)
 
Theoremtrubifal 1573 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.)
((⊤ ↔ ⊥) ↔ ⊥)
 
Theoremfalbifal 1574 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ↔ ⊥) ↔ ⊤)
 
1.2.16.4  Conjunction
 
Theoremtruantru 1575 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∧ ⊤) ↔ ⊤)
 
Theoremtruanfal 1576 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∧ ⊥) ↔ ⊥)
 
Theoremfalantru 1577 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∧ ⊤) ↔ ⊥)
 
Theoremfalanfal 1578 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∧ ⊥) ↔ ⊥)
 
1.2.16.5  Disjunction
 
Theoremtruortru 1579 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ∨ ⊤) ↔ ⊤)
 
Theoremtruorfal 1580 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∨ ⊥) ↔ ⊤)
 
Theoremfalortru 1581 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∨ ⊤) ↔ ⊤)
 
Theoremfalorfal 1582 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ∨ ⊥) ↔ ⊥)
 
1.2.16.6  Alternative denial
 
Theoremtrunantru 1583 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ⊼ ⊤) ↔ ⊥)
 
Theoremtrunanfal 1584 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.)
((⊤ ⊼ ⊥) ↔ ⊤)
 
Theoremfalnantru 1585 A identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ⊼ ⊤) ↔ ⊤)
 
Theoremfalnanfal 1586 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ⊼ ⊥) ↔ ⊤)
 
1.2.16.7  Exclusive disjunction
 
Theoremtruxortru 1587 A identity. (Contributed by David A. Wheeler, 8-May-2015.)
((⊤ ⊻ ⊤) ↔ ⊥)
 
Theoremtruxorfal 1588 A identity. (Contributed by David A. Wheeler, 8-May-2015.)
((⊤ ⊻ ⊥) ↔ ⊤)
 
Theoremfalxortru 1589 A identity. (Contributed by David A. Wheeler, 9-May-2015.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
((⊥ ⊻ ⊤) ↔ ⊤)
 
Theoremfalxorfal 1590 A identity. (Contributed by David A. Wheeler, 9-May-2015.)
((⊥ ⊻ ⊥) ↔ ⊥)
 
1.2.16.8  Joint denial
 
Theoremtrunortru 1591 A identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 7-Dec-2023.)
((⊤ ⊤) ↔ ⊥)
 
TheoremtrunortruOLD 1592 Obsolete version of trunortru 1591 as of 7-Dec-2023. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ⊤) ↔ ⊥)
 
Theoremtrunorfal 1593 A identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
((⊤ ⊥) ↔ ⊥)
 
TheoremtrunorfalOLD 1594 Obsolete version of trunorfal 1593 as of 17-Dec-2023. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ⊥) ↔ ⊥)
 
Theoremfalnortru 1595 A identity. (Contributed by Remi, 25-Oct-2023.)
((⊥ ⊤) ↔ ⊥)
 
Theoremfalnorfal 1596 A identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
((⊥ ⊥) ↔ ⊤)
 
TheoremfalnorfalOLD 1597 Obsolete version of falnorfal 1596 as of 17-Dec-2023. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊥ ⊥) ↔ ⊤)
 
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 1507) and the logical conjunction (df-an 397).

The full adder takes into account an "input carry", so it has three inputs and again two outputs, corresponding to the "sum" (df-had 1599) and "updated carry" (df-cad 1613).

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
 
Syntaxwhad 1598 Syntax for the "sum" output of the full adder. (Contributed by Mario Carneiro, 4-Sep-2016.)
wff hadd(𝜑, 𝜓, 𝜒)
 
Definitiondf-had 1599 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) ↔ ((𝜑𝜓) ⊻ 𝜒))
 
Theoremhadbi123d 1600 Equality theorem for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → (𝜂𝜁))       (𝜑 → (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁)))
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