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Theorem List for Metamath Proof Explorer - 1501-1600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnanbi1d 1501 Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))
 
Theoremnanbi2d 1502 Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))
 
Theoremnanbi12d 1503 Join two logical equivalences with anti-conjunction. (Contributed by Scott Fenton, 2-Jan-2018.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
1.2.12  Logical 'xor'
 
Syntaxwxo 1504 Extend wff definition to include exclusive disjunction ('xor').
wff (𝜑𝜓)
 
Definitiondf-xor 1505 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 1526) and the constant false (df-fal 1529), we will be able to prove these truth table values: ((⊤ ⊻ ⊤) ↔ ⊥) (truxortru 1568), ((⊤ ⊻ ⊥) ↔ ⊤) (truxorfal 1569), ((⊥ ⊻ ⊤) ↔ ⊤) (falxortru 1570), and ((⊥ ⊻ ⊥) ↔ ⊥) (falxorfal 1571). Contrast with (df-an 385), (df-or 384), (wi 4), and (df-nan 1488) . (Contributed by FL, 22-Nov-2010.)
((𝜑𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremxnor 1506 Two ways to write XNOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremxorcom 1507 The connector is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓) ↔ (𝜓𝜑))
 
Theoremxorass 1508 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 1509 This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.)
((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
 
Theoremxor2 1510 Two ways to express "exclusive or." (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
 
Theoremxoror 1511 XOR implies OR. (Contributed by BJ, 19-Apr-2019.)
((𝜑𝜓) → (𝜑𝜓))
 
Theoremxornan 1512 XOR implies NAND. (Contributed by BJ, 19-Apr-2019.)
((𝜑𝜓) → ¬ (𝜑𝜓))
 
Theoremxornan2 1513 XOR implies NAND (written with the connector). (Contributed by BJ, 19-Apr-2019.)
((𝜑𝜓) → (𝜑𝜓))
 
Theoremxorneg2 1514 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 1515 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 1516 The connector is unchanged under negation of both arguments. (Contributed by Mario Carneiro, 4-Sep-2016.)
((¬ 𝜑 ⊻ ¬ 𝜓) ↔ (𝜑𝜓))
 
Theoremxorbi12i 1517 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))
 
Theoremxorbi12d 1518 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theoremanxordi 1519 Conjunction distributes over exclusive-or. In intuitionistic logic this assertion is also true, even though xordi 955 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 1520 Exclusive-or variant of the law of the excluded middle (exmid 430). 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)
 
1.2.13  True and false constants
 
1.2.13.1  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 1526 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in definition df-ex 1745 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 1531 may be adopted and this subsection moved down to the start of the subsection with wex 1744 below. However, the use of dftru2 1531 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

 
Syntaxwal 1521 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
 
1.2.13.2  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 1526 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in theorem equs3 1932 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 1531 may be adopted and this subsection moved down to just above weq 1931 below. However, the use of dftru2 1531 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

 
Syntaxcv 1522 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 2637. Since (when 𝑦 is distinct from 𝑥) we have 𝑥 = {𝑦𝑦𝑥} by cvjust 2646, 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 1522 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1522 is intrinsically no different from any other class-building syntax such as cab 2637, cun 3605, or c0 3948.

For a general discussion of the theory of classes and the role of cv 1522, 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 1931 of predicate calculus from the wceq 1523 of set theory, so that we don't overload the = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.)

class 𝑥
 
Syntaxwceq 1523 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 1931 of predicate calculus in terms of the wceq 1523 of set theory, so that we don't "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 1931 or wceq 1523, 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 2644 for more information on the set theory usage of wceq 1523.)

wff 𝐴 = 𝐵
 
1.2.13.3  Define the true and false constants
 
Syntaxwtru 1524 The constant is a wff.
wff
 
Theoremtrujust 1525 Soundness justification theorem for df-tru 1526. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.)
((∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥) ↔ (∀𝑦 𝑦 = 𝑦 → ∀𝑦 𝑦 = 𝑦))
 
Definitiondf-tru 1526 Definition of the truth value "true", or "verum", denoted by . This is a tautology, as proved by tru 1527. 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 id 22 instance 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 1527, and other proofs should depend on tru 1527 (directly or indirectly) 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 1527 instead. (New usage is discouraged.)
(⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
 
Theoremtru 1527 The truth value is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
 
Syntaxwfal 1528 The constant is a wff.
wff
 
Definitiondf-fal 1529 Definition of the truth value "false", or "falsum", denoted by . See also df-tru 1526. (Contributed by Anthony Hart, 22-Oct-2010.)
(⊥ ↔ ¬ ⊤)
 
Theoremfal 1530 The truth value is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
¬ ⊥
 
Theoremdftru2 1531 An alternate definition of "true". (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) (New usage is discouraged.)
(⊤ ↔ (𝜑𝜑))
 
Theoremtrut 1532 A proposition is equivalent to it being implied by . Closed form of trud 1533. Dual of dfnot 1542. It is to tbtru 1534 what a1bi 351 is to tbt 358. (Contributed by BJ, 26-Oct-2019.)
(𝜑 ↔ (⊤ → 𝜑))
 
Theoremtrud 1533 Eliminate as an antecedent. A proposition implied by is true. (Contributed by Mario Carneiro, 13-Mar-2014.)
(⊤ → 𝜑)       𝜑
 
Theoremtbtru 1534 A proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.)
(𝜑 ↔ (𝜑 ↔ ⊤))
 
Theoremnbfal 1535 The negation of a proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.)
𝜑 ↔ (𝜑 ↔ ⊥))
 
Theorembitru 1536 A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
𝜑       (𝜑 ↔ ⊤)
 
Theorembifal 1537 A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
¬ 𝜑       (𝜑 ↔ ⊥)
 
Theoremfalim 1538 The truth value implies anything. Also called the "principle of explosion", or "ex falso [sequitur]] quodlibet" (Latin for "from falsehood, anything [follows]]"). (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
(⊥ → 𝜑)
 
Theoremfalimd 1539 The truth value implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑 ∧ ⊥) → 𝜓)
 
Theorema1tru 1540 Anything implies . (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
(𝜑 → ⊤)
 
Theoremtruan 1541 True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
((⊤ ∧ 𝜑) ↔ 𝜑)
 
Theoremdfnot 1542 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 1543 Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑𝜓) → ⊥)       (𝜑 → ¬ 𝜓)
 
Theoremefald 1544 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑 ∧ ¬ 𝜓) → ⊥)       (𝜑𝜓)
 
Theorempm2.21fal 1545 If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜓)       (𝜑 → ⊥)
 
1.2.14  Truth tables

Some sources define operations on true/false values using truth tables. These tables show the results of their operations for all possible combinations of true () and false (). Here we show that our definitions and axioms produce equivalent results for (conjunction aka logical 'and') df-an 385, (disjunction aka logical inclusive 'or') df-or 384, (implies) wi 4, ¬ (not) wn 3, (logical equivalence) df-bi 197, (nand aka Sheffer stroke) df-nan 1488, and (exclusive or) df-xor 1505.

 
Theoremtruantru 1546 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∧ ⊤) ↔ ⊤)
 
Theoremtruanfal 1547 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∧ ⊥) ↔ ⊥)
 
Theoremfalantru 1548 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∧ ⊤) ↔ ⊥)
 
Theoremfalanfal 1549 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∧ ⊥) ↔ ⊥)
 
Theoremtruortru 1550 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ∨ ⊤) ↔ ⊤)
 
Theoremtruorfal 1551 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∨ ⊥) ↔ ⊤)
 
Theoremfalortru 1552 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∨ ⊤) ↔ ⊤)
 
Theoremfalorfal 1553 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ∨ ⊥) ↔ ⊥)
 
Theoremtruimtru 1554 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ → ⊤) ↔ ⊤)
 
Theoremtruimfal 1555 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ → ⊥) ↔ ⊥)
 
Theoremfalimtru 1556 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ → ⊤) ↔ ⊤)
 
Theoremfalimfal 1557 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ → ⊥) ↔ ⊤)
 
Theoremnottru 1558 A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(¬ ⊤ ↔ ⊥)
 
Theoremnotfal 1559 A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(¬ ⊥ ↔ ⊤)
 
Theoremtrubitru 1560 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ↔ ⊤) ↔ ⊤)
 
Theoremfalbitru 1561 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 1562 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 1563 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ↔ ⊥) ↔ ⊤)
 
Theoremtrunantru 1564 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ⊼ ⊤) ↔ ⊥)
 
Theoremtrunanfal 1565 A identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
((⊤ ⊼ ⊥) ↔ ⊤)
 
Theoremfalnantru 1566 A identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ⊼ ⊤) ↔ ⊤)
 
Theoremfalnanfal 1567 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ⊼ ⊥) ↔ ⊤)
 
Theoremtruxortru 1568 A identity. (Contributed by David A. Wheeler, 8-May-2015.)
((⊤ ⊻ ⊤) ↔ ⊥)
 
Theoremtruxorfal 1569 A identity. (Contributed by David A. Wheeler, 8-May-2015.)
((⊤ ⊻ ⊥) ↔ ⊤)
 
Theoremfalxortru 1570 A identity. (Contributed by David A. Wheeler, 9-May-2015.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
((⊥ ⊻ ⊤) ↔ ⊤)
 
Theoremfalxorfal 1571 A identity. (Contributed by David A. Wheeler, 9-May-2015.)
((⊥ ⊻ ⊥) ↔ ⊥)
 
1.2.15  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 1505) and the logical conjunction (df-an 385).

The full adder takes into account an "input carry", so it has three inputs and again two outputs, corresponding to the "sum" (df-had 1573) and "updated carry" (df-cad 1586). 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.

 
1.2.15.1  Full adder: sum
 
Syntaxwhad 1572 Syntax for the "sum" output of the full adder. (Contributed by Mario Carneiro, 4-Sep-2016.)
wff hadd(𝜑, 𝜓, 𝜒)
 
Definitiondf-had 1573 Definition of the "sum" output of the full adder (triple exclusive disjunction, or XOR3). (Contributed by Mario Carneiro, 4-Sep-2016.)
(hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ⊻ 𝜒))
 
Theoremhadbi123d 1574 Equality theorem for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → (𝜂𝜁))       (𝜑 → (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁)))
 
Theoremhadbi123i 1575 Equality theorem for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       (hadd(𝜑, 𝜒, 𝜏) ↔ hadd(𝜓, 𝜃, 𝜂))
 
Theoremhadass 1576 Associative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
(hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓𝜒)))
 
Theoremhadbi 1577 The adder sum is the same as the triple biconditional. (Contributed by Mario Carneiro, 4-Sep-2016.)
(hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ↔ 𝜒))
 
Theoremhadcoma 1578 Commutative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
(hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜑, 𝜒))
 
Theoremhadcomb 1579 Commutative law for the adders sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
(hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜑, 𝜒, 𝜓))
 
Theoremhadrot 1580 Rotation law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
(hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜒, 𝜑))
 
Theoremhadnot 1581 The adder sum distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)
(¬ hadd(𝜑, 𝜓, 𝜒) ↔ hadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒))
 
Theoremhad1 1582 If the first input is true, then the adder sum is equivalent to the biconditionality of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)
(𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓𝜒)))
 
Theoremhad0 1583 If the first input is false, then the adder sum is equivalent to the exclusive disjunction of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jul-2020.)
𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓𝜒)))
 
Theoremhadifp 1584 The value of the adder sum is, if the first input is true, the biconditionality, and if the first input is false, the exclusive disjunction, of the other two inputs. (Contributed by BJ, 11-Aug-2020.)
(hadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓𝜒), (𝜓𝜒)))
 
1.2.15.2  Full adder: carry
 
Syntaxwcad 1585 Syntax for the "carry" output of the full adder. (Contributed by Mario Carneiro, 4-Sep-2016.)
wff cadd(𝜑, 𝜓, 𝜒)
 
Definitiondf-cad 1586 Definition of the "carry" output of the full adder. It is true when at least two arguments are true, so it is equal to the "majority" function on three variables. See cador 1587 and cadan 1588 for alternate definitions. (Contributed by Mario Carneiro, 4-Sep-2016.)
(cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (𝜒 ∧ (𝜑𝜓))))
 
Theoremcador 1587 The adder carry in disjunctive normal form. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)
(cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜓𝜒)))
 
Theoremcadan 1588 The adder carry in conjunctive normal form. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 25-Sep-2018.)
(cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜓𝜒)))
 
Theoremcadbi123d 1589 Equality theorem for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → (𝜂𝜁))       (𝜑 → (cadd(𝜓, 𝜃, 𝜂) ↔ cadd(𝜒, 𝜏, 𝜁)))
 
Theoremcadbi123i 1590 Equality theorem for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       (cadd(𝜑, 𝜒, 𝜏) ↔ cadd(𝜓, 𝜃, 𝜂))
 
Theoremcadcoma 1591 Commutative law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
(cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜓, 𝜑, 𝜒))
 
Theoremcadcomb 1592 Commutative law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)
(cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜑, 𝜒, 𝜓))
 
Theoremcadrot 1593 Rotation law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
(cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜓, 𝜒, 𝜑))
 
Theoremcadnot 1594 The adder carry distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)
(¬ cadd(𝜑, 𝜓, 𝜒) ↔ cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒))
 
Theoremcad1 1595 If one input is true, then the adder carry is true exactly when at least one of the other two inputs is true. (Contributed by Mario Carneiro, 8-Sep-2016.) (Proof shortened by Wolf Lammen, 19-Jun-2020.)
(𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑𝜓)))
 
Theoremcad0 1596 If one input is false, then the adder carry is true exactly when both of the other two inputs are true. (Contributed by Mario Carneiro, 8-Sep-2016.)
𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑𝜓)))
 
Theoremcadifp 1597 The value of the carry is, if the input carry is true, the disjunction, and if the input carry is false, the conjunction, of the other two inputs. (Contributed by BJ, 8-Oct-2019.)
(cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜒, (𝜑𝜓), (𝜑𝜓)))
 
Theoremcad11 1598 If (at least) two inputs are true, then the adder carry is true. (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓) → cadd(𝜑, 𝜓, 𝜒))
 
Theoremcadtru 1599 The adder carry is true as soon as its first two inputs are the truth constant. (Contributed by Mario Carneiro, 4-Sep-2016.)
cadd(⊤, ⊤, 𝜑)
 
1.3  Other axiomatizations related to classical propositional calculus
 
1.3.1  Minimal implicational calculus

Minimal implicational calculus, or intuitionistic implicational calculus, or positive implicational calculus, is the implicational fragment of minimal calculus (which is also the implicational fragment of intuitionistic calculus and of positive calculus). It is sometimes called "C-pure intuitionism" since the letter C is sometimes used to denote implication, especially in prefix notation. It can be axiomatized by the inference rule of modus ponens ax-mp 5 together with the axioms { ax-1 6, ax-2 7 } (sometimes written KS), or with { imim1 83, ax-1 6, pm2.43 56 } (written B'KW), or with { imim2 58, pm2.04 90, ax-1 6, pm2.43 56 } (written BCKW), or with the single axiom minimp 1600. This section proves minimp 1600 from { ax-1 6, ax-2 7 }, and then the converse, due to Ivo Thomas.

Sources for this section are the webpage https://web.ics.purdue.edu/~dulrich/C-pure-intuitionism-page.htm on Ted Ulrich's website, and the articles C. A. Meredith, A single axiom of positive logic, Journal of computing systems, vol. 1 (1953), 169--170, and C. A. Meredith, A. N. Prior, Notes on the axiomatics of the propositional calculus, Notre Dame Journal of Formal Logic, vol. 4 (1963), 171--187.

We may use a compact notation for derivations known as the D-notation where "D" stands for "condensed Detachment". For instance, "D21" means detaching ax-1 6 from ax-2 7, that is, using modus ponens ax-mp 5 with ax-1 6 as minor premise and ax-2 7 as major premise. D-strings are accepted by the grammar Dstr := digit | "D" Dstr Dstr.

(Contributed by BJ, 11-Apr-2021.)

 
Theoremminimp 1600 A single axiom for minimal implicational calculus, due to Meredith. Other single axioms of the same length are known, but it is thought to be the minimal length. (Contributed by BJ, 4-Apr-2021.)
(𝜑 → ((𝜓𝜒) → (((𝜃𝜓) → (𝜒𝜏)) → (𝜓𝜏))))
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