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Theorem List for Metamath Proof Explorer - 1601-1700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3anandis 1601 Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.)
(((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜃)) → 𝜏)       ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)
 
Theorem3anandirs 1602 Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.)
(((𝜑𝜃) ∧ (𝜓𝜃) ∧ (𝜒𝜃)) → 𝜏)       (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)
 
Theoremecase23d 1603 Deduction for elimination by cases. (Contributed by NM, 22-Apr-1994.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → ¬ 𝜃)    &   (𝜑 → (𝜓𝜒𝜃))       (𝜑𝜓)
 
Theorem3ecase 1604 Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.)
𝜑𝜃)    &   𝜓𝜃)    &   𝜒𝜃)    &   ((𝜑𝜓𝜒) → 𝜃)       𝜃
 
Theorem3bior1fd 1605 A disjunction is equivalent to a threefold disjunction with single falsehood, analogous to biorf 967. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
(𝜑 → ¬ 𝜃)       (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜒𝜓)))
 
Theorem3bior1fand 1606 A disjunction is equivalent to a threefold disjunction with single falsehood of a conjunction. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
(𝜑 → ¬ 𝜃)       (𝜑 → ((𝜒𝜓) ↔ ((𝜃𝜏) ∨ 𝜒𝜓)))
 
Theorem3bior2fd 1607 A wff is equivalent to its threefold disjunction with double falsehood, analogous to biorf 967. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
(𝜑 → ¬ 𝜃)    &   (𝜑 → ¬ 𝜒)       (𝜑 → (𝜓 ↔ (𝜃𝜒𝜓)))
 
Theorem3biant1d 1608 A conjunction is equivalent to a threefold conjunction with single truth, analogous to biantrud 529. (Contributed by Alexander van der Vekens, 26-Sep-2017.)
(𝜑𝜃)       (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜒𝜓)))
 
Theoremintn3an1d 1609 Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ (𝜓𝜒𝜃))
 
Theoremintn3an2d 1610 Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ (𝜒𝜓𝜃))
 
Theoremintn3an3d 1611 Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ (𝜒𝜃𝜓))
 
Theoreman3andi 1612 Distribution of conjunction over threefold conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.)
((𝜑 ∧ (𝜓𝜒𝜃)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜃)))
 
Theoreman33rean 1613 Rearrange a 9-fold conjunction. (Contributed by Thierry Arnoux, 14-Apr-2019.)
(((𝜑𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎𝜌)) ↔ ((𝜑𝜏𝜌) ∧ ((𝜓𝜃) ∧ (𝜂𝜎) ∧ (𝜒𝜁))))
 
1.2.12  Logical "nand" (Sheffer stroke)
 
Syntaxwnan 1614 Extend wff definition to include alternative denial ("nand").
wff (𝜑𝜓)
 
Definitiondf-nan 1615 Define incompatibility, or alternative denial ("not-and" or "nand"). This is also called the Sheffer stroke, represented by a vertical bar, but we use a different symbol to avoid ambiguity with other uses of the vertical bar. In the second edition of Principia Mathematica (1927), Russell and Whitehead used the Sheffer stroke and suggested it as a replacement for the "or" and "not" operations of the first edition. However, in practice, "or" and "not" are more widely used. After we define the constant true (df-tru 1662) and the constant false (df-fal 1672), we will be able to prove these truth table values: ((⊤ ⊼ ⊤) ↔ ⊥) (trunantru 1700), ((⊤ ⊼ ⊥) ↔ ⊤) (trunanfal 1701), ((⊥ ⊼ ⊤) ↔ ⊤) (falnantru 1702), and ((⊥ ⊼ ⊥) ↔ ⊤) (falnanfal 1703). Contrast with (df-an 387), (df-or 881), (wi 4), and (df-xor 1640). (Contributed by Jeff Hoffman, 19-Nov-2007.)
((𝜑𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremnanan 1616 Conjunction in terms of alternative denial. (Contributed by Mario Carneiro, 9-May-2015.)
((𝜑𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremnanimn 1617 Alternative denial in terms of our primitive connectives (implication and negation). (Contributed by WL, 26-Jun-2020.)
((𝜑𝜓) ↔ (𝜑 → ¬ 𝜓))
 
Theoremnanor 1618 Alternative denial in terms of disjunction and negation. This explains the name "alternative denial". (Contributed by BJ, 19-Oct-2022.)
((𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
 
Theoremnancom 1619 Alternative denial is commutative. Remark: alternative denial is not associative, see nanass 1637. (Contributed by Mario Carneiro, 9-May-2015.) (Proof shortened by Wolf Lammen, 26-Jun-2020.)
((𝜑𝜓) ↔ (𝜓𝜑))
 
TheoremnancomOLD 1620 Obsolete proof of nancom 1619 as of 19-Oct-2022. (Contributed by Mario Carneiro, 9-May-2015.) (Proof shortened by Wolf Lammen, 7-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ↔ (𝜓𝜑))
 
Theoremnannan 1621 Nested alternative denials. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 26-Jun-2020.)
((𝜑 ⊼ (𝜓𝜒)) ↔ (𝜑 → (𝜓𝜒)))
 
TheoremnannanOLD 1622 Obsolete proof of nannan 1621 as of 19-Oct-2022. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 7-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ⊼ (𝜓𝜒)) ↔ (𝜑 → (𝜓𝜒)))
 
Theoremnanim 1623 Implication in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007.)
((𝜑𝜓) ↔ (𝜑 ⊼ (𝜓𝜓)))
 
Theoremnannot 1624 Negation in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Revised by Wolf Lammen, 26-Jun-2020.)
𝜑 ↔ (𝜑𝜑))
 
TheoremnannotOLD 1625 Obsolete proof of nannot 1624 as of 19-Oct-2022. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜓 ↔ (𝜓𝜓))
 
Theoremnanbi 1626 Biconditional in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 27-Jun-2020.)
((𝜑𝜓) ↔ ((𝜑𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜓𝜓))))
 
Theoremnanbi1 1627 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.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))
 
Theoremnanbi1OLD 1628 Obsolete proof of nanbi1 1627 as of 19-Oct-2022. (Contributed by Anthony Hart, 1-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))
 
Theoremnanbi2 1629 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.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))
 
Theoremnanbi12 1630 Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) ↔ (𝜓𝜃)))
 
Theoremnanbi1i 1631 Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜓𝜒))
 
Theoremnanbi2i 1632 Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
(𝜑𝜓)       ((𝜒𝜑) ↔ (𝜒𝜓))
 
Theoremnanbi12i 1633 Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))
 
Theoremnanbi1d 1634 Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))
 
Theoremnanbi2d 1635 Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))
 
Theoremnanbi12d 1636 Join two logical equivalences with anti-conjunction. (Contributed by Scott Fenton, 2-Jan-2018.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theoremnanass 1637 A characterization of when an expression involving alternative denials associates. Remark: alternative denial is commutative, see nancom 1619. (Contributed by Richard Penner, 29-Feb-2020.) (Proof shortened by Wolf Lammen, 23-Oct-2022.)
((𝜑𝜒) ↔ (((𝜑𝜓) ⊼ 𝜒) ↔ (𝜑 ⊼ (𝜓𝜒))))
 
TheoremnanassOLD 1638 Obsolete proof of nanass 1637 as of 23-Oct-2022. (Contributed by Richard Penner, 29-Feb-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜒) ↔ (((𝜑𝜓) ⊼ 𝜒) ↔ (𝜑 ⊼ (𝜓𝜒))))
 
1.2.13  Logical "xor"
 
Syntaxwxo 1639 Extend wff definition to include exclusive disjunction ("xor").
wff (𝜑𝜓)
 
Definitiondf-xor 1640 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 1662) and the constant false (df-fal 1672), we will be able to prove these truth table values: ((⊤ ⊻ ⊤) ↔ ⊥) (truxortru 1704), ((⊤ ⊻ ⊥) ↔ ⊤) (truxorfal 1705), ((⊥ ⊻ ⊤) ↔ ⊤) (falxortru 1706), and ((⊥ ⊻ ⊥) ↔ ⊥) (falxorfal 1707). Contrast with (df-an 387), (df-or 881), (wi 4), and (df-nan 1615). (Contributed by FL, 22-Nov-2010.)
((𝜑𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremxnor 1641 Two ways to write XNOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremxorcom 1642 The connector is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓) ↔ (𝜓𝜑))
 
Theoremxorass 1643 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 1644 This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.)
((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
 
Theoremxor2 1645 Two ways to express "exclusive or". (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
 
Theoremxoror 1646 XOR implies OR. (Contributed by BJ, 19-Apr-2019.)
((𝜑𝜓) → (𝜑𝜓))
 
Theoremxornan 1647 XOR implies NAND. (Contributed by BJ, 19-Apr-2019.)
((𝜑𝜓) → ¬ (𝜑𝜓))
 
Theoremxornan2 1648 XOR implies NAND (written with the connector). (Contributed by BJ, 19-Apr-2019.)
((𝜑𝜓) → (𝜑𝜓))
 
Theoremxorneg2 1649 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 1650 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 1651 The connector is unchanged under negation of both arguments. (Contributed by Mario Carneiro, 4-Sep-2016.)
((¬ 𝜑 ⊻ ¬ 𝜓) ↔ (𝜑𝜓))
 
Theoremxorbi12i 1652 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))
 
Theoremxorbi12d 1653 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theoremanxordi 1654 Conjunction distributes over exclusive-or. In intuitionistic logic this assertion is also true, even though xordi 1047 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 1655 Exclusive-or variant of the law of the excluded middle (exmid 925). 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  True and false constants
 
1.2.14.1  Universal quantifier for use by df-tru

Even though it isn't ordinarily part of propositional calculus, the universal quantifier is introduced here so that the soundness of definition df-tru 1662 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in definition df-ex 1881 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 1664 may be adopted and this subsection moved down to the start of the subsection with wex 1880 below. However, the use of dftru2 1664 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

 
Syntaxwal 1656 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
 
1.2.14.2  Equality predicate for use by df-tru

Even though it isn't ordinarily part of propositional calculus, the equality predicate = is introduced here so that the soundness of definition df-tru 1662 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in theorem equs3 2064 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 1664 may be adopted and this subsection moved down to just above weq 2063 below. However, the use of dftru2 1664 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

 
Syntaxcv 1657 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 2811. Since (when 𝑦 is distinct from 𝑥) we have 𝑥 = {𝑦𝑦𝑥} by cvjust 2820, 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 1657 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1657 is intrinsically no different from any other class-building syntax such as cab 2811, cun 3796, or c0 4144.

For a general discussion of the theory of classes and the role of cv 1657, see mmset.html#class.

(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 2063 of predicate calculus from the wceq 1658 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 1658 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 2063 of predicate calculus in terms of the wceq 1658 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 2063 or wceq 1658, 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 2818 for more information on the set theory usage of wceq 1658.)

wff 𝐴 = 𝐵
 
1.2.14.3  The true constant
 
Syntaxwtru 1659 The constant is a wff.
wff
 
Theoremtrujust 1660 Soundness justification theorem for df-tru 1662. Instance of monothetic 258. (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))
 
TheoremtrujustOLD 1661 Obsolete proof of trujust 1660 as of 7-Sep-2022. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥) ↔ (∀𝑦 𝑦 = 𝑦 → ∀𝑦 𝑦 = 𝑦))
 
Definitiondf-tru 1662 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 1663, and other proofs should use tru 1663 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 1663 instead. (New usage is discouraged.)
(⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
 
Theoremtru 1663 The truth value is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
 
Theoremdftru2 1664 An alternate definition of "true" (see comment of df-tru 1662). The associated justification theorem is monothetic 258. (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) Use tru 1663 instead. (New usage is discouraged.)
(⊤ ↔ (𝜑𝜑))
 
Theoremtrut 1665 A proposition is equivalent to it being implied by . Closed form of mptru 1666. Dual of dfnot 1678. It is to tbtru 1667 what a1bi 354 is to tbt 361. (Contributed by BJ, 26-Oct-2019.)
(𝜑 ↔ (⊤ → 𝜑))
 
Theoremmptru 1666 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 1663). (Contributed by Mario Carneiro, 13-Mar-2014.)
(⊤ → 𝜑)       𝜑
 
Theoremtbtru 1667 A proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.)
(𝜑 ↔ (𝜑 ↔ ⊤))
 
Theorembitru 1668 A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
𝜑       (𝜑 ↔ ⊤)
 
Theoremtrud 1669 Anything implies . Dual statement of falim 1676. Deduction form of tru 1663. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
(𝜑 → ⊤)
 
Theoremtruan 1670 True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
((⊤ ∧ 𝜑) ↔ 𝜑)
 
1.2.14.4  The false constant
 
Syntaxwfal 1671 The constant is a wff.
wff
 
Definitiondf-fal 1672 Definition of the truth value "false", or "falsum", denoted by . See also df-tru 1662. (Contributed by Anthony Hart, 22-Oct-2010.)
(⊥ ↔ ¬ ⊤)
 
Theoremfal 1673 The truth value is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
¬ ⊥
 
Theoremnbfal 1674 The negation of a proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.)
𝜑 ↔ (𝜑 ↔ ⊥))
 
Theorembifal 1675 A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
¬ 𝜑       (𝜑 ↔ ⊥)
 
Theoremfalim 1676 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 1669. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
(⊥ → 𝜑)
 
Theoremfalimd 1677 The truth value implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑 ∧ ⊥) → 𝜓)
 
Theoremdfnot 1678 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 1679 Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑𝜓) → ⊥)       (𝜑 → ¬ 𝜓)
 
Theoremefald 1680 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑 ∧ ¬ 𝜓) → ⊥)       (𝜑𝜓)
 
Theorempm2.21fal 1681 If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜓)       (𝜑 → ⊥)
 
1.2.15  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 199, conjunction df-an 387, disjunction df-or 881, alternative denial df-nan 1615, exclusive disjunction df-xor 1640.

 
1.2.15.1  Implication
 
Theoremtruimtru 1682 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using trud 1669 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 1683 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ → ⊥) ↔ ⊥)
 
Theoremfalimtru 1684 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using falim 1676 instead of trud 1669 but the present proof using trud 1669 emphasizes that the result does not require the principle of explosion. (Proof modification is discouraged.)
((⊥ → ⊤) ↔ ⊤)
 
Theoremfalimfal 1685 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using falim 1676 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.)
((⊥ → ⊥) ↔ ⊤)
 
1.2.15.2  Negation
 
Theoremnottru 1686 A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(¬ ⊤ ↔ ⊥)
 
Theoremnotfal 1687 A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(¬ ⊥ ↔ ⊤)
 
1.2.15.3  Equivalence
 
Theoremtrubitru 1688 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ↔ ⊤) ↔ ⊤)
 
Theoremfalbitru 1689 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 1690 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 1691 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ↔ ⊥) ↔ ⊤)
 
1.2.15.4  Conjunction
 
Theoremtruantru 1692 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∧ ⊤) ↔ ⊤)
 
Theoremtruanfal 1693 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∧ ⊥) ↔ ⊥)
 
Theoremfalantru 1694 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∧ ⊤) ↔ ⊥)
 
Theoremfalanfal 1695 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∧ ⊥) ↔ ⊥)
 
1.2.15.5  Disjunction
 
Theoremtruortru 1696 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ∨ ⊤) ↔ ⊤)
 
Theoremtruorfal 1697 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∨ ⊥) ↔ ⊤)
 
Theoremfalortru 1698 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∨ ⊤) ↔ ⊤)
 
Theoremfalorfal 1699 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ∨ ⊥) ↔ ⊥)
 
1.2.15.6  Alternative denial
 
Theoremtrunantru 1700 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ⊼ ⊤) ↔ ⊥)
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