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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 3anandis 1601 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.) |
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃)) → 𝜏) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏) | ||
Theorem | 3anandirs 1602 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.) |
⊢ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜃) ∧ (𝜒 ∧ 𝜃)) → 𝜏) ⇒ ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) | ||
Theorem | ecase23d 1603 | Deduction for elimination by cases. (Contributed by NM, 22-Apr-1994.) |
⊢ (𝜑 → ¬ 𝜒) & ⊢ (𝜑 → ¬ 𝜃) & ⊢ (𝜑 → (𝜓 ∨ 𝜒 ∨ 𝜃)) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | 3ecase 1604 | Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.) |
⊢ (¬ 𝜑 → 𝜃) & ⊢ (¬ 𝜓 → 𝜃) & ⊢ (¬ 𝜒 → 𝜃) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ 𝜃 | ||
Theorem | 3bior1fd 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 → ((𝜒 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒 ∨ 𝜓))) | ||
Theorem | 3bior1fand 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 → ((𝜒 ∨ 𝜓) ↔ ((𝜃 ∧ 𝜏) ∨ 𝜒 ∨ 𝜓))) | ||
Theorem | 3bior2fd 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 ↔ (𝜃 ∨ 𝜒 ∨ 𝜓))) | ||
Theorem | 3biant1d 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.) |
⊢ (𝜑 → 𝜃) ⇒ ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜒 ∧ 𝜓))) | ||
Theorem | intn3an1d 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 → ¬ (𝜓 ∧ 𝜒 ∧ 𝜃)) | ||
Theorem | intn3an2d 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 → ¬ (𝜒 ∧ 𝜓 ∧ 𝜃)) | ||
Theorem | intn3an3d 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 → ¬ (𝜒 ∧ 𝜃 ∧ 𝜓)) | ||
Theorem | an3andi 1612 | Distribution of conjunction over threefold conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.) |
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒) ∧ (𝜑 ∧ 𝜃))) | ||
Theorem | an33rean 1613 | Rearrange a 9-fold conjunction. (Contributed by Thierry Arnoux, 14-Apr-2019.) |
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏 ∧ 𝜂) ∧ (𝜁 ∧ 𝜎 ∧ 𝜌)) ↔ ((𝜑 ∧ 𝜏 ∧ 𝜌) ∧ ((𝜓 ∧ 𝜃) ∧ (𝜂 ∧ 𝜎) ∧ (𝜒 ∧ 𝜁)))) | ||
Syntax | wnan 1614 | Extend wff definition to include alternative denial ("nand"). |
wff (𝜑 ⊼ 𝜓) | ||
Definition | df-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.) |
⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) | ||
Theorem | nanan 1616 | Conjunction in terms of alternative denial. (Contributed by Mario Carneiro, 9-May-2015.) |
⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ⊼ 𝜓)) | ||
Theorem | nanimn 1617 | Alternative denial in terms of our primitive connectives (implication and negation). (Contributed by WL, 26-Jun-2020.) |
⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜑 → ¬ 𝜓)) | ||
Theorem | nanor 1618 | Alternative denial in terms of disjunction and negation. This explains the name "alternative denial". (Contributed by BJ, 19-Oct-2022.) |
⊢ ((𝜑 ⊼ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)) | ||
Theorem | nancom 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.) |
⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜓 ⊼ 𝜑)) | ||
Theorem | nancomOLD 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.) |
⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜓 ⊼ 𝜑)) | ||
Theorem | nannan 1621 | Nested alternative denials. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 26-Jun-2020.) |
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒))) | ||
Theorem | nannanOLD 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 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒))) | ||
Theorem | nanim 1623 | Implication in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007.) |
⊢ ((𝜑 → 𝜓) ↔ (𝜑 ⊼ (𝜓 ⊼ 𝜓))) | ||
Theorem | nannot 1624 | Negation in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Revised by Wolf Lammen, 26-Jun-2020.) |
⊢ (¬ 𝜑 ↔ (𝜑 ⊼ 𝜑)) | ||
Theorem | nannotOLD 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 ↔ (𝜓 ⊼ 𝜓)) | ||
Theorem | nanbi 1626 | Biconditional in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 27-Jun-2020.) |
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓)))) | ||
Theorem | nanbi1 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.) |
⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜒))) | ||
Theorem | nanbi1OLD 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.) |
⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜒))) | ||
Theorem | nanbi2 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.) |
⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ⊼ 𝜑) ↔ (𝜒 ⊼ 𝜓))) | ||
Theorem | nanbi12 1630 | Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.) |
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜃))) | ||
Theorem | nanbi1i 1631 | Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜒)) | ||
Theorem | nanbi2i 1632 | 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 1633 | Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ↔ 𝜃) ⇒ ⊢ ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜃)) | ||
Theorem | nanbi1d 1634 | 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 1635 | 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 1636 | 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 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.) |
⊢ ((𝜑 ↔ 𝜒) ↔ (((𝜑 ⊼ 𝜓) ⊼ 𝜒) ↔ (𝜑 ⊼ (𝜓 ⊼ 𝜒)))) | ||
Theorem | nanassOLD 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.) |
⊢ ((𝜑 ↔ 𝜒) ↔ (((𝜑 ⊼ 𝜓) ⊼ 𝜒) ↔ (𝜑 ⊼ (𝜓 ⊼ 𝜒)))) | ||
Syntax | wxo 1639 | Extend wff definition to include exclusive disjunction ("xor"). |
wff (𝜑 ⊻ 𝜓) | ||
Definition | df-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.) |
⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)) | ||
Theorem | xnor 1641 | Two ways to write XNOR. (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓)) | ||
Theorem | xorcom 1642 | The connector ⊻ is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑)) | ||
Theorem | xorass 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.) |
⊢ (((𝜑 ⊻ 𝜓) ⊻ 𝜒) ↔ (𝜑 ⊻ (𝜓 ⊻ 𝜒))) | ||
Theorem | excxor 1644 | This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.) |
⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓))) | ||
Theorem | xor2 1645 | Two ways to express "exclusive or". (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))) | ||
Theorem | xoror 1646 | XOR implies OR. (Contributed by BJ, 19-Apr-2019.) |
⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ∨ 𝜓)) | ||
Theorem | xornan 1647 | XOR implies NAND. (Contributed by BJ, 19-Apr-2019.) |
⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ∧ 𝜓)) | ||
Theorem | xornan2 1648 | XOR implies NAND (written with the ⊼ connector). (Contributed by BJ, 19-Apr-2019.) |
⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ⊼ 𝜓)) | ||
Theorem | xorneg2 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) ↔ ¬ (𝜑 ⊻ 𝜓)) | ||
Theorem | xorneg1 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.) |
⊢ ((¬ 𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓)) | ||
Theorem | xorneg 1651 | The connector ⊻ is unchanged under negation of both arguments. (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ ((¬ 𝜑 ⊻ ¬ 𝜓) ↔ (𝜑 ⊻ 𝜓)) | ||
Theorem | xorbi12i 1652 | Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ↔ 𝜃) ⇒ ⊢ ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃)) | ||
Theorem | xorbi12d 1653 | Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → (𝜃 ↔ 𝜏)) ⇒ ⊢ (𝜑 → ((𝜓 ⊻ 𝜃) ↔ (𝜒 ⊻ 𝜏))) | ||
Theorem | anxordi 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 ∧ (𝜓 ⊻ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒))) | ||
Theorem | xorexmid 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) | ||
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. | ||
Syntax | wal 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 | ||
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. | ||
Syntax | cv 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 {𝑦 ∣ 𝑦 ∈ 𝑥} is a class by cab 2811.
Since (when
𝑦 is distinct from 𝑥) we
have 𝑥 =
{𝑦 ∣ 𝑦 ∈ 𝑥} by
cvjust 2820, 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 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 | ||
Syntax | wceq 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 | ||
Syntax | wtru 1659 | The constant ⊤ is a wff. |
wff ⊤ | ||
Theorem | trujust 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)) | ||
Theorem | trujustOLD 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)) | ||
Definition | df-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)) | ||
Theorem | tru 1663 | The truth value ⊤ is provable. (Contributed by Anthony Hart, 13-Oct-2010.) |
⊢ ⊤ | ||
Theorem | dftru2 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.) |
⊢ (⊤ ↔ (𝜑 → 𝜑)) | ||
Theorem | trut 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)) | ||
Theorem | mptru 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 | ||
Theorem | tbtru 1667 | A proposition is equivalent to itself being equivalent to ⊤. (Contributed by Anthony Hart, 14-Aug-2011.) |
⊢ (𝜑 ↔ (𝜑 ↔ ⊤)) | ||
Theorem | bitru 1668 | A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.) |
⊢ 𝜑 ⇒ ⊢ (𝜑 ↔ ⊤) | ||
Theorem | trud 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 → ⊤) | ||
Theorem | truan 1670 | True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.) |
⊢ ((⊤ ∧ 𝜑) ↔ 𝜑) | ||
Syntax | wfal 1671 | The constant ⊥ is a wff. |
wff ⊥ | ||
Definition | df-fal 1672 | Definition of the truth value "false", or "falsum", denoted by ⊥. See also df-tru 1662. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (⊥ ↔ ¬ ⊤) | ||
Theorem | fal 1673 | The truth value ⊥ is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.) |
⊢ ¬ ⊥ | ||
Theorem | nbfal 1674 | The negation of a proposition is equivalent to itself being equivalent to ⊥. (Contributed by Anthony Hart, 14-Aug-2011.) |
⊢ (¬ 𝜑 ↔ (𝜑 ↔ ⊥)) | ||
Theorem | bifal 1675 | A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.) |
⊢ ¬ 𝜑 ⇒ ⊢ (𝜑 ↔ ⊥) | ||
Theorem | falim 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) | ||
Theorem | falimd 1677 | The truth value ⊥ implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ ((𝜑 ∧ ⊥) → 𝜓) | ||
Theorem | dfnot 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 → ⊥)) | ||
Theorem | inegd 1679 | Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ ((𝜑 ∧ 𝜓) → ⊥) ⇒ ⊢ (𝜑 → ¬ 𝜓) | ||
Theorem | efald 1680 | Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ ((𝜑 ∧ ¬ 𝜓) → ⊥) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | pm2.21fal 1681 | If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → ¬ 𝜓) ⇒ ⊢ (𝜑 → ⊥) | ||
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. | ||
Theorem | truimtru 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.) |
⊢ ((⊤ → ⊤) ↔ ⊤) | ||
Theorem | truimfal 1683 | A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊤ → ⊥) ↔ ⊥) | ||
Theorem | falimtru 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.) |
⊢ ((⊥ → ⊤) ↔ ⊤) | ||
Theorem | falimfal 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.) |
⊢ ((⊥ → ⊥) ↔ ⊤) | ||
Theorem | nottru 1686 | A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (¬ ⊤ ↔ ⊥) | ||
Theorem | notfal 1687 | A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (¬ ⊥ ↔ ⊤) | ||
Theorem | trubitru 1688 | A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊤ ↔ ⊤) ↔ ⊤) | ||
Theorem | falbitru 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.) |
⊢ ((⊥ ↔ ⊤) ↔ ⊥) | ||
Theorem | trubifal 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.) |
⊢ ((⊤ ↔ ⊥) ↔ ⊥) | ||
Theorem | falbifal 1691 | A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊥ ↔ ⊥) ↔ ⊤) | ||
Theorem | truantru 1692 | A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊤ ∧ ⊤) ↔ ⊤) | ||
Theorem | truanfal 1693 | A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊤ ∧ ⊥) ↔ ⊥) | ||
Theorem | falantru 1694 | A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊥ ∧ ⊤) ↔ ⊥) | ||
Theorem | falanfal 1695 | A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊥ ∧ ⊥) ↔ ⊥) | ||
Theorem | truortru 1696 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊤ ∨ ⊤) ↔ ⊤) | ||
Theorem | truorfal 1697 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊤ ∨ ⊥) ↔ ⊤) | ||
Theorem | falortru 1698 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊥ ∨ ⊤) ↔ ⊤) | ||
Theorem | falorfal 1699 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊥ ∨ ⊥) ↔ ⊥) | ||
Theorem | trunantru 1700 | A ⊼ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊤ ⊼ ⊤) ↔ ⊥) |
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