HomeHome Metamath Proof Explorer
Theorem List (p. 16 of 466)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-29289)
  Hilbert Space Explorer  Hilbert Space Explorer
(29290-30812)
  Users' Mathboxes  Users' Mathboxes
(30813-46532)
 

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 1542) and the constant false (df-fal 1552), we will be able to prove these truth table values: ((⊤ ⊻ ⊤) ↔ ⊥) (truxortru 1584), ((⊤ ⊻ ⊥) ↔ ⊤) (truxorfal 1585), ((⊥ ⊻ ⊤) ↔ ⊤) (falxortru 1586), and ((⊥ ⊻ ⊥) ↔ ⊥) (falxorfal 1587). 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 1542) and the constant false (df-fal 1552), we will be able to prove these truth table values: ((⊤ ⊤) ↔ ⊥) (trunortru 1588), ((⊤ ⊥) ↔ ⊥) (trunorfal 1589), ((⊥ ⊤) ↔ ⊥) (falnortru 1591), and ((⊥ ⊥) ↔ ⊤) (falnorfal 1592). 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))
 
Theoremnoran 1530 is expressible via . (Contributed by Remi, 26-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
((𝜑𝜓) ↔ ((𝜑 𝜑) (𝜓 𝜓)))
 
Theoremnoror 1531 is expressible via . (Contributed by Remi, 26-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
((𝜑𝜓) ↔ ((𝜑 𝜓) (𝜑 𝜓)))
 
Theoremnorasslem1 1532 This lemma shows the equivalence of two expressions, used in norass 1535. (Contributed by Wolf Lammen, 18-Dec-2023.)
(((𝜑𝜓) → 𝜒) ↔ ((𝜑 𝜓) ∨ 𝜒))
 
Theoremnorasslem2 1533 This lemma specializes biimt 361 suitably for the proof of norass 1535. (Contributed by Wolf Lammen, 18-Dec-2023.)
(𝜑 → (𝜓 ↔ ((𝜑𝜒) → 𝜓)))
 
Theoremnorasslem3 1534 This lemma specializes biorf 934 suitably for the proof of norass 1535. (Contributed by Wolf Lammen, 18-Dec-2023.)
𝜑 → ((𝜓𝜒) ↔ ((𝜑𝜓) → 𝜒)))
 
Theoremnorass 1535 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 1536 Obsolete version of norass 1535 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 1542 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in Definition df-ex 1783 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 1544 may be adopted and this subsection moved down to the start of the subsection with wex 1782 below. However, the use of dftru2 1544 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

 
Syntaxwal 1537 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 1542 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in Theorem weq 1967 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 1544 may be adopted and this subsection moved down to just above weq 1967 below. However, the use of dftru2 1544 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

 
Syntaxcv 1538 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 2716. Since (when 𝑦 is distinct from 𝑥) we have 𝑥 = {𝑦𝑦𝑥} by cvjust 2733, 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 1538 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1538 is intrinsically no different from any other class-building syntax such as cab 2716, cun 3886, or c0 4257.

For a general discussion of the theory of classes and the role of cv 1538, see mmset.html#class 1538.

(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 1967 of predicate calculus from the wceq 1539 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 1539 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 1967 of predicate calculus in terms of the wceq 1539 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 1967 or wceq 1539, 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 2731 for more information on the set theory usage of wceq 1539.)

wff 𝐴 = 𝐵
 
1.2.15.3  The true constant
 
Syntaxwtru 1540 The constant is a wff.
wff
 
Theoremtrujust 1541 Soundness justification theorem for df-tru 1542. 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 1542 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 1543, and other proofs should use tru 1543 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 1543 instead. (New usage is discouraged.)
(⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
 
Theoremtru 1543 The truth value is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
 
Theoremdftru2 1544 An alternate definition of "true" (see comment of df-tru 1542). The associated justification theorem is monothetic 265. (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) Use tru 1543 instead. (New usage is discouraged.)
(⊤ ↔ (𝜑𝜑))
 
Theoremtrut 1545 A proposition is equivalent to it being implied by . Closed form of mptru 1546. Dual of dfnot 1558. It is to tbtru 1547 what a1bi 363 is to tbt 370. (Contributed by BJ, 26-Oct-2019.)
(𝜑 ↔ (⊤ → 𝜑))
 
Theoremmptru 1546 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 1543). (Contributed by Mario Carneiro, 13-Mar-2014.)
(⊤ → 𝜑)       𝜑
 
Theoremtbtru 1547 A proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.)
(𝜑 ↔ (𝜑 ↔ ⊤))
 
Theorembitru 1548 A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
𝜑       (𝜑 ↔ ⊤)
 
Theoremtrud 1549 Anything implies . Dual statement of falim 1556. Deduction form of tru 1543. Note on naming: in 2022, the theorem now known as mptru 1546 was renamed from trud so if you are reading documentation written before that time, references to trud refer to what is now mptru 1546. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
(𝜑 → ⊤)
 
Theoremtruan 1550 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 1551 The constant is a wff.
wff
 
Definitiondf-fal 1552 Definition of the truth value "false", or "falsum", denoted by . See also df-tru 1542. (Contributed by Anthony Hart, 22-Oct-2010.)
(⊥ ↔ ¬ ⊤)
 
Theoremfal 1553 The truth value is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
¬ ⊥
 
Theoremnbfal 1554 The negation of a proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.)
𝜑 ↔ (𝜑 ↔ ⊥))
 
Theorembifal 1555 A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
¬ 𝜑       (𝜑 ↔ ⊥)
 
Theoremfalim 1556 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 1549. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
(⊥ → 𝜑)
 
Theoremfalimd 1557 The truth value implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑 ∧ ⊥) → 𝜓)
 
Theoremdfnot 1558 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 1559 Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑𝜓) → ⊥)       (𝜑 → ¬ 𝜓)
 
Theoremefald 1560 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑 ∧ ¬ 𝜓) → ⊥)       (𝜑𝜓)
 
Theorempm2.21fal 1561 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 1562 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using trud 1549 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 1563 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ → ⊥) ↔ ⊥)
 
Theoremfalimtru 1564 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using falim 1556 instead of trud 1549 but the present proof using trud 1549 emphasizes that the result does not require the principle of explosion. (Proof modification is discouraged.)
((⊥ → ⊤) ↔ ⊤)
 
Theoremfalimfal 1565 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using falim 1556 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 1566 A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(¬ ⊤ ↔ ⊥)
 
Theoremnotfal 1567 A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(¬ ⊥ ↔ ⊤)
 
1.2.16.3  Equivalence
 
Theoremtrubitru 1568 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ↔ ⊤) ↔ ⊤)
 
Theoremfalbitru 1569 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 1570 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 1571 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ↔ ⊥) ↔ ⊤)
 
1.2.16.4  Conjunction
 
Theoremtruantru 1572 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∧ ⊤) ↔ ⊤)
 
Theoremtruanfal 1573 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∧ ⊥) ↔ ⊥)
 
Theoremfalantru 1574 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∧ ⊤) ↔ ⊥)
 
Theoremfalanfal 1575 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∧ ⊥) ↔ ⊥)
 
1.2.16.5  Disjunction
 
Theoremtruortru 1576 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ∨ ⊤) ↔ ⊤)
 
Theoremtruorfal 1577 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∨ ⊥) ↔ ⊤)
 
Theoremfalortru 1578 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∨ ⊤) ↔ ⊤)
 
Theoremfalorfal 1579 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ∨ ⊥) ↔ ⊥)
 
1.2.16.6  Alternative denial
 
Theoremtrunantru 1580 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ⊼ ⊤) ↔ ⊥)
 
Theoremtrunanfal 1581 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 1582 A identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ⊼ ⊤) ↔ ⊤)
 
Theoremfalnanfal 1583 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ⊼ ⊥) ↔ ⊤)
 
1.2.16.7  Exclusive disjunction
 
Theoremtruxortru 1584 A identity. (Contributed by David A. Wheeler, 8-May-2015.)
((⊤ ⊻ ⊤) ↔ ⊥)
 
Theoremtruxorfal 1585 A identity. (Contributed by David A. Wheeler, 8-May-2015.)
((⊤ ⊻ ⊥) ↔ ⊤)
 
Theoremfalxortru 1586 A identity. (Contributed by David A. Wheeler, 9-May-2015.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
((⊥ ⊻ ⊤) ↔ ⊤)
 
Theoremfalxorfal 1587 A identity. (Contributed by David A. Wheeler, 9-May-2015.)
((⊥ ⊻ ⊥) ↔ ⊥)
 
1.2.16.8  Joint denial
 
Theoremtrunortru 1588 A identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 7-Dec-2023.)
((⊤ ⊤) ↔ ⊥)
 
Theoremtrunorfal 1589 A identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
((⊤ ⊥) ↔ ⊥)
 
TheoremtrunorfalOLD 1590 Obsolete version of trunorfal 1589 as of 17-Dec-2023. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ⊥) ↔ ⊥)
 
Theoremfalnortru 1591 A identity. (Contributed by Remi, 25-Oct-2023.)
((⊥ ⊤) ↔ ⊥)
 
Theoremfalnorfal 1592 A identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
((⊥ ⊥) ↔ ⊤)
 
TheoremfalnorfalOLD 1593 Obsolete version of falnorfal 1592 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 1595) and "updated carry" (df-cad 1609).

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 1594 Syntax for the "sum" output of the full adder. (Contributed by Mario Carneiro, 4-Sep-2016.)
wff hadd(𝜑, 𝜓, 𝜒)
 
Definitiondf-had 1595 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 1596 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)))
 
Theoremhadbi123i 1597 Equality theorem for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       (hadd(𝜑, 𝜒, 𝜏) ↔ hadd(𝜓, 𝜃, 𝜂))
 
Theoremhadass 1598 Associative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
(hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓𝜒)))
 
Theoremhadbi 1599 The adder sum is the same as the triple biconditional. (Contributed by Mario Carneiro, 4-Sep-2016.)
(hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ↔ 𝜒))
 
Theoremhadcoma 1600 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))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46532
  Copyright terms: Public domain < Previous  Next >