P. 16 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 1501-1600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nanbi12i 1501	Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜃))
Theorem	nanbi1d 1502	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 1503	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 1504	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 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.)
⊢ ((𝜑 ↔ 𝜒) ↔ (((𝜑 ⊼ 𝜓) ⊼ 𝜒) ↔ (𝜑 ⊼ (𝜓 ⊼ 𝜒))))
1.2.13  Logical "xor"
Syntax	wxo 1506	Extend wff definition to include exclusive disjunction ("xor").
wff (𝜑 ⊻ 𝜓)
Definition	df-xor 1507	Define exclusive disjunction (logical "xor"). Return true if either the left or right, but not both, are true. After we define the constant true ⊤ (df-tru 1545) and the constant false ⊥ (df-fal 1555), we will be able to prove these truth table values: ((⊤ ⊻ ⊤) ↔ ⊥) (truxortru 1587), ((⊤ ⊻ ⊥) ↔ ⊤) (truxorfal 1588), ((⊥ ⊻ ⊤) ↔ ⊤) (falxortru 1589), and ((⊥ ⊻ ⊥) ↔ ⊥) (falxorfal 1590). Contrast with ∧ (df-an 397), ∨ (df-or 845), → (wi 4), and ⊼ (df-nan 1487). (Contributed by FL, 22-Nov-2010.)
⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
Theorem	xnor 1508	Two ways to write XNOR (exclusive not-or). (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓))
Theorem	xorcom 1509	The connector ⊻ is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 21-Apr-2024.)
⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑))
Theorem	xorcomOLD 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.)
⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑))
Theorem	xorass 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.)
⊢ (((𝜑 ⊻ 𝜓) ⊻ 𝜒) ↔ (𝜑 ⊻ (𝜓 ⊻ 𝜒)))
Theorem	excxor 1512	This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.)
⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓)))
Theorem	xor2 1513	Two ways to express "exclusive or". (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
Theorem	xoror 1514	Exclusive disjunction implies disjunction ("XOR implies OR"). (Contributed by BJ, 19-Apr-2019.)
⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ∨ 𝜓))
Theorem	xornan 1515	Exclusive disjunction implies alternative denial ("XOR implies NAND"). (Contributed by BJ, 19-Apr-2019.)
⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ∧ 𝜓))
Theorem	xornan2 1516	XOR implies NAND (written with the ⊼ connector). (Contributed by BJ, 19-Apr-2019.)
⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ⊼ 𝜓))
Theorem	xorneg2 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) ↔ ¬ (𝜑 ⊻ 𝜓))
Theorem	xorneg1 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.)
⊢ ((¬ 𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓))
Theorem	xorneg 1519	The connector ⊻ is unchanged under negation of both arguments. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ ((¬ 𝜑 ⊻ ¬ 𝜓) ↔ (𝜑 ⊻ 𝜓))
Theorem	xorbi12i 1520	Equality property for exclusive disjunction. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 21-Apr-2024.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃))
Theorem	xorbi12iOLD 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.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃))
Theorem	xorbi12d 1522	Equality property for exclusive disjunction. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 ⊻ 𝜃) ↔ (𝜒 ⊻ 𝜏)))
Theorem	anxordi 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 ∧ (𝜓 ⊻ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒)))
Theorem	xorexmid 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"
Syntax	wnor 1525	Extend wff definition to include joint denial ("nor").
wff (𝜑 ⊽ 𝜓)
Definition	df-nor 1526	Define joint denial ("not-or" or "nor"). After we define the constant true ⊤ (df-tru 1545) and the constant false ⊥ (df-fal 1555), we will be able to prove these truth table values: ((⊤ ⊽ ⊤) ↔ ⊥) (trunortru 1591), ((⊤ ⊽ ⊥) ↔ ⊥) (trunorfal 1593), ((⊥ ⊽ ⊤) ↔ ⊥) (falnortru 1595), and ((⊥ ⊽ ⊥) ↔ ⊤) (falnorfal 1596). Contrast with ∧ (df-an 397), ∨ (df-or 845), → (wi 4), ⊼ (df-nan 1487), and ⊻ (df-xor 1507). (Contributed by Remi, 25-Oct-2023.)
⊢ ((𝜑 ⊽ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓))
Theorem	norcom 1527	The connector ⊽ is commutative. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 23-Apr-2024.)
⊢ ((𝜑 ⊽ 𝜓) ↔ (𝜓 ⊽ 𝜑))
Theorem	norcomOLD 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))
Theorem	nornot 1529	¬ is expressible via ⊽. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
⊢ (¬ 𝜑 ↔ (𝜑 ⊽ 𝜑))
Theorem	nornotOLD 1530	Obsolete version of nornot 1529 as of 8-Dec-2023. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ 𝜑 ↔ (𝜑 ⊽ 𝜑))
Theorem	noran 1531	∧ is expressible via ⊽. (Contributed by Remi, 26-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
⊢ ((𝜑 ∧ 𝜓) ↔ ((𝜑 ⊽ 𝜑) ⊽ (𝜓 ⊽ 𝜓)))
Theorem	noranOLD 1532	Obsolete version of noran 1531 as of 8-Dec-2023. (Contributed by Remi, 26-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓) ↔ ((𝜑 ⊽ 𝜑) ⊽ (𝜓 ⊽ 𝜓)))
Theorem	noror 1533	∨ is expressible via ⊽. (Contributed by Remi, 26-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
⊢ ((𝜑 ∨ 𝜓) ↔ ((𝜑 ⊽ 𝜓) ⊽ (𝜑 ⊽ 𝜓)))
Theorem	nororOLD 1534	Obsolete version of noror 1533 as of 8-Dec-2023. (Contributed by Remi, 26-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∨ 𝜓) ↔ ((𝜑 ⊽ 𝜓) ⊽ (𝜑 ⊽ 𝜓)))
Theorem	norasslem1 1535	This lemma shows the equivalence of two expressions, used in norass 1538. (Contributed by Wolf Lammen, 18-Dec-2023.)
⊢ (((𝜑 ∨ 𝜓) → 𝜒) ↔ ((𝜑 ⊽ 𝜓) ∨ 𝜒))
Theorem	norasslem2 1536	This lemma specializes biimt 361 suitably for the proof of norass 1538. (Contributed by Wolf Lammen, 18-Dec-2023.)
⊢ (𝜑 → (𝜓 ↔ ((𝜑 ∨ 𝜒) → 𝜓)))
Theorem	norasslem3 1537	This lemma specializes biorf 934 suitably for the proof of norass 1538. (Contributed by Wolf Lammen, 18-Dec-2023.)
⊢ (¬ 𝜑 → ((𝜓 → 𝜒) ↔ ((𝜑 ∨ 𝜓) → 𝜒)))
Theorem	norass 1538	A characterization of when an expression involving joint denials associates. This is identical to the case when alternative denial is associative, see nanass 1505. Remark: Like alternative denial, joint denial is also commutative, see norcom 1527. (Contributed by RP, 29-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
⊢ ((𝜑 ↔ 𝜒) ↔ (((𝜑 ⊽ 𝜓) ⊽ 𝜒) ↔ (𝜑 ⊽ (𝜓 ⊽ 𝜒))))
Theorem	norassOLD 1539	Obsolete version of norass 1538 as of 17-Dec-2023. (Contributed by RP, 29-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ↔ 𝜒) ↔ (((𝜑 ⊽ 𝜓) ⊽ 𝜒) ↔ (𝜑 ⊽ (𝜓 ⊽ 𝜒))))
1.2.15  True and false constants
1.2.15.1  Universal quantifier for use by df-tru Even though it is not ordinarily part of propositional calculus, the universal quantifier ∀ is introduced here so that the soundness of Definition df-tru 1545 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in Definition df-ex 1787 in the predicate calculus section below. For those who want propositional calculus to be self-contained, i.e., to use wff variables only, the alternate Definition dftru2 1547 may be adopted and this subsection moved down to the start of the subsection with wex 1786 below. However, the use of dftru2 1547 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.
Syntax	wal 1540	Extend wff definition to include the universal quantifier ("for all"). ∀𝑥𝜑 is read "𝜑 (phi) is true for all 𝑥". Typically, in its final application 𝜑 would be replaced with a wff containing a (free) occurrence of the variable 𝑥, for example 𝑥 = 𝑦. In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of 𝑥. When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same.
wff ∀𝑥𝜑
1.2.15.2  Equality predicate for use by df-tru Even though it is not ordinarily part of propositional calculus, the equality predicate = is introduced here so that the soundness of definition df-tru 1545 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in Theorem weq 1970 in the predicate calculus section below. For those who want propositional calculus to be self-contained, i.e., to use wff variables only, the alternate definition dftru2 1547 may be adopted and this subsection moved down to just above weq 1970 below. However, the use of dftru2 1547 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.
Syntax	cv 1541	This syntax construction states that a variable 𝑥, which has been declared to be a setvar variable by $f statement vx, is also a class expression. This can be justified informally as follows. We know that the class builder {𝑦 ∣ 𝑦 ∈ 𝑥} is a class by cab 2717. Since (when 𝑦 is distinct from 𝑥) we have 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} by cvjust 2734, 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 1541 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1541 is intrinsically no different from any other class-building syntax such as cab 2717, cun 3890, or c0 4262. For a general discussion of the theory of classes and the role of cv 1541, see mmset.html#class 1541. (The description above applies to set theory, not predicate calculus. The purpose of introducing class 𝑥 here, and not in set theory where it belongs, is to allow us to express, i.e., "prove", the weq 1970 of predicate calculus from the wceq 1542 of set theory, so that we do not overload the = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.)
class 𝑥
Syntax	wceq 1542	Extend wff definition to include class equality. For a general discussion of the theory of classes, see mmset.html#class. (The purpose of introducing wff 𝐴 = 𝐵 here, and not in set theory where it belongs, is to allow us to express, i.e., "prove", the weq 1970 of predicate calculus in terms of the wceq 1542 of set theory, so that we do not "overload" the = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. For example, some parsers - although not the Metamath program - stumble on the fact that the = in 𝑥 = 𝑦 could be the = of either weq 1970 or wceq 1542, although mathematically it makes no difference. The class variables 𝐴 and 𝐵 are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 2732 for more information on the set theory usage of wceq 1542.)
wff 𝐴 = 𝐵
1.2.15.3  The true constant
Syntax	wtru 1543	The constant ⊤ is a wff.
wff ⊤
Theorem	trujust 1544	Soundness justification theorem for df-tru 1545. Instance of monothetic 265. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.)
⊢ ((∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥) ↔ (∀𝑦 𝑦 = 𝑦 → ∀𝑦 𝑦 = 𝑦))
Definition	df-tru 1545	Definition of the truth value "true", or "verum", denoted by ⊤. In this definition, an instance of id 22 is used as the definiens, although any tautology, such as an axiom, can be used in its place. This particular instance of id 22 was chosen so this definition can be checked by the same algorithm that is used for predicate calculus. This definition should be referenced directly only by tru 1546, and other proofs should use tru 1546 instead of this definition, since there are many alternate ways to define ⊤. (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by NM, 11-Jul-2019.) Use tru 1546 instead. (New usage is discouraged.)
⊢ (⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
Theorem	tru 1546	The truth value ⊤ is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
⊢ ⊤
Theorem	dftru2 1547	An alternate definition of "true" (see comment of df-tru 1545). The associated justification theorem is monothetic 265. (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) Use tru 1546 instead. (New usage is discouraged.)
⊢ (⊤ ↔ (𝜑 → 𝜑))
Theorem	trut 1548	A proposition is equivalent to it being implied by ⊤. Closed form of mptru 1549. Dual of dfnot 1561. It is to tbtru 1550 what a1bi 363 is to tbt 370. (Contributed by BJ, 26-Oct-2019.)
⊢ (𝜑 ↔ (⊤ → 𝜑))
Theorem	mptru 1549	Eliminate ⊤ as an antecedent. A proposition implied by ⊤ is true. This is modus ponens ax-mp 5 when the minor hypothesis is ⊤ (which holds by tru 1546). (Contributed by Mario Carneiro, 13-Mar-2014.)
⊢ (⊤ → 𝜑)    ⇒   ⊢ 𝜑
Theorem	tbtru 1550	A proposition is equivalent to itself being equivalent to ⊤. (Contributed by Anthony Hart, 14-Aug-2011.)
⊢ (𝜑 ↔ (𝜑 ↔ ⊤))
Theorem	bitru 1551	A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
⊢ 𝜑    ⇒   ⊢ (𝜑 ↔ ⊤)
Theorem	trud 1552	Anything implies ⊤. Dual statement of falim 1559. Deduction form of tru 1546. Note on naming: in 2022, the theorem now known as mptru 1549 was renamed from trud so if you are reading documentation written before that time, references to trud refer to what is now mptru 1549. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
⊢ (𝜑 → ⊤)
Theorem	truan 1553	True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
⊢ ((⊤ ∧ 𝜑) ↔ 𝜑)
1.2.15.4  The false constant
Syntax	wfal 1554	The constant ⊥ is a wff.
wff ⊥
Definition	df-fal 1555	Definition of the truth value "false", or "falsum", denoted by ⊥. See also df-tru 1545. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ (⊥ ↔ ¬ ⊤)
Theorem	fal 1556	The truth value ⊥ is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
⊢ ¬ ⊥
Theorem	nbfal 1557	The negation of a proposition is equivalent to itself being equivalent to ⊥. (Contributed by Anthony Hart, 14-Aug-2011.)
⊢ (¬ 𝜑 ↔ (𝜑 ↔ ⊥))
Theorem	bifal 1558	A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
⊢ ¬ 𝜑    ⇒   ⊢ (𝜑 ↔ ⊥)
Theorem	falim 1559	The truth value ⊥ implies anything. Also called the "principle of explosion", or "ex falso [sequitur]] quodlibet" (Latin for "from falsehood, anything [follows]]"). Dual statement of trud 1552. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
⊢ (⊥ → 𝜑)
Theorem	falimd 1560	The truth value ⊥ implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ ((𝜑 ∧ ⊥) → 𝜓)
Theorem	dfnot 1561	Given falsum ⊥, we can define the negation of a wff 𝜑 as the statement that ⊥ follows from assuming 𝜑. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
⊢ (¬ 𝜑 ↔ (𝜑 → ⊥))
Theorem	inegd 1562	Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ ((𝜑 ∧ 𝜓) → ⊥)    ⇒   ⊢ (𝜑 → ¬ 𝜓)
Theorem	efald 1563	Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ ((𝜑 ∧ ¬ 𝜓) → ⊥)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	pm2.21fal 1564	If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ⊥)
1.2.16  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
Theorem	truimtru 1565	A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using trud 1552 instead of id 22 but the principle of identity id 22 is more basic, and the present proof indicates that the result still holds in relevance logic. (Proof modification is discouraged.)
⊢ ((⊤ → ⊤) ↔ ⊤)
Theorem	truimfal 1566	A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊤ → ⊥) ↔ ⊥)
Theorem	falimtru 1567	A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using falim 1559 instead of trud 1552 but the present proof using trud 1552 emphasizes that the result does not require the principle of explosion. (Proof modification is discouraged.)
⊢ ((⊥ → ⊤) ↔ ⊤)
Theorem	falimfal 1568	A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) An alternate proof is possible using falim 1559 instead of id 22 but the present proof using id 22 emphasizes that the result does not require the principle of explosion. (Proof modification is discouraged.)
⊢ ((⊥ → ⊥) ↔ ⊤)
1.2.16.2  Negation
Theorem	nottru 1569	A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ (¬ ⊤ ↔ ⊥)
Theorem	notfal 1570	A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ (¬ ⊥ ↔ ⊤)
1.2.16.3  Equivalence
Theorem	trubitru 1571	A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊤ ↔ ⊤) ↔ ⊤)
Theorem	falbitru 1572	A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
⊢ ((⊥ ↔ ⊤) ↔ ⊥)
Theorem	trubifal 1573	A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
⊢ ((⊤ ↔ ⊥) ↔ ⊥)
Theorem	falbifal 1574	A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊥ ↔ ⊥) ↔ ⊤)
1.2.16.4  Conjunction
Theorem	truantru 1575	A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊤ ∧ ⊤) ↔ ⊤)
Theorem	truanfal 1576	A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊤ ∧ ⊥) ↔ ⊥)
Theorem	falantru 1577	A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊥ ∧ ⊤) ↔ ⊥)
Theorem	falanfal 1578	A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊥ ∧ ⊥) ↔ ⊥)
1.2.16.5  Disjunction
Theorem	truortru 1579	A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊤ ∨ ⊤) ↔ ⊤)
Theorem	truorfal 1580	A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊤ ∨ ⊥) ↔ ⊤)
Theorem	falortru 1581	A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
⊢ ((⊥ ∨ ⊤) ↔ ⊤)
Theorem	falorfal 1582	A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊥ ∨ ⊥) ↔ ⊥)
1.2.16.6  Alternative denial
Theorem	trunantru 1583	A ⊼ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊤ ⊼ ⊤) ↔ ⊥)
Theorem	trunanfal 1584	A ⊼ identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
⊢ ((⊤ ⊼ ⊥) ↔ ⊤)
Theorem	falnantru 1585	A ⊼ identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊥ ⊼ ⊤) ↔ ⊤)
Theorem	falnanfal 1586	A ⊼ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊥ ⊼ ⊥) ↔ ⊤)
1.2.16.7  Exclusive disjunction
Theorem	truxortru 1587	A ⊻ identity. (Contributed by David A. Wheeler, 8-May-2015.)
⊢ ((⊤ ⊻ ⊤) ↔ ⊥)
Theorem	truxorfal 1588	A ⊻ identity. (Contributed by David A. Wheeler, 8-May-2015.)
⊢ ((⊤ ⊻ ⊥) ↔ ⊤)
Theorem	falxortru 1589	A ⊻ identity. (Contributed by David A. Wheeler, 9-May-2015.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
⊢ ((⊥ ⊻ ⊤) ↔ ⊤)
Theorem	falxorfal 1590	A ⊻ identity. (Contributed by David A. Wheeler, 9-May-2015.)
⊢ ((⊥ ⊻ ⊥) ↔ ⊥)
1.2.16.8  Joint denial
Theorem	trunortru 1591	A ⊽ identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 7-Dec-2023.)
⊢ ((⊤ ⊽ ⊤) ↔ ⊥)
Theorem	trunortruOLD 1592	Obsolete version of trunortru 1591 as of 7-Dec-2023. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((⊤ ⊽ ⊤) ↔ ⊥)
Theorem	trunorfal 1593	A ⊽ identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
⊢ ((⊤ ⊽ ⊥) ↔ ⊥)
Theorem	trunorfalOLD 1594	Obsolete version of trunorfal 1593 as of 17-Dec-2023. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((⊤ ⊽ ⊥) ↔ ⊥)
Theorem	falnortru 1595	A ⊽ identity. (Contributed by Remi, 25-Oct-2023.)
⊢ ((⊥ ⊽ ⊤) ↔ ⊥)
Theorem	falnorfal 1596	A ⊽ identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
⊢ ((⊥ ⊽ ⊥) ↔ ⊤)
Theorem	falnorfalOLD 1597	Obsolete version of falnorfal 1596 as of 17-Dec-2023. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((⊥ ⊽ ⊥) ↔ ⊤)
1.2.17  Half adder and full adder in propositional calculus Propositional calculus deals with truth values, which can be interpreted as bits. Using this, we can define the half adder and the full adder in pure propositional calculus, and show their basic properties. The half adder adds two 1-bit numbers. Its two outputs are the "sum" S and the "carry" C. The real sum is then given by 2C+S. The sum and carry correspond respectively to the logical exclusive disjunction (df-xor 1507) and the logical conjunction (df-an 397). The full adder takes into account an "input carry", so it has three inputs and again two outputs, corresponding to the "sum" (df-had 1599) and "updated carry" (df-cad 1613). Here is a short description. We code the bit 0 by ⊥ and 1 by ⊤. Even though hadd and cadd are invariant under permutation of their arguments, assume for the sake of concreteness that 𝜑 (resp. 𝜓) is the i^th bit of the first (resp. second) number to add (with the convention that the i^th bit is the multiple of 2^i in the base-2 representation), and that 𝜒 is the i^th carry (with the convention that the 0^th carry is 0). Then, hadd(𝜑, 𝜓, 𝜒) gives the i^th bit of the sum, and cadd(𝜑, 𝜓, 𝜒) gives the (i+1)^th carry. Then, addition is performed by iteration from i = 0 to i = 1 + (max of the number of digits of the two summands) by "updating" the carry.
1.2.17.1  Full adder: sum
Syntax	whad 1598	Syntax for the "sum" output of the full adder. (Contributed by Mario Carneiro, 4-Sep-2016.)
wff hadd(𝜑, 𝜓, 𝜒)
Definition	df-had 1599	Definition of the "sum" output of the full adder (triple exclusive disjunction, or XOR3, or testing whether an odd number of parameters are true). (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ⊻ 𝜓) ⊻ 𝜒))
Theorem	hadbi123d 1600	Equality theorem for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    &   ⊢ (𝜑 → (𝜂 ↔ 𝜁))    ⇒   ⊢ (𝜑 → (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁)))