P. 17 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 1601-1700   *Has distinct variable group(s)

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))))
1.2.12  Logical "nand" (Sheffer stroke)
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))))
1.2.13  Logical "xor"
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)
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.
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
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.
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
1.2.14.3  The true constant
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)
1.2.14.4  The false constant
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 → ⊥)
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
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.)
⊢ ((⊥ → ⊥) ↔ ⊤)
1.2.15.2  Negation
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.)
⊢ (¬ ⊥ ↔ ⊤)
1.2.15.3  Equivalence
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.)
⊢ ((⊥ ↔ ⊥) ↔ ⊤)
1.2.15.4  Conjunction
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.)
⊢ ((⊥ ∧ ⊥) ↔ ⊥)
1.2.15.5  Disjunction
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.)
⊢ ((⊥ ∨ ⊥) ↔ ⊥)
1.2.15.6  Alternative denial
Theorem	trunantru 1700	A ⊼ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊤ ⊼ ⊤) ↔ ⊥)