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

Theoreminegd 1501 Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑𝜓) → ⊥)       (𝜑 → ¬ 𝜓)

Theoremefald 1502 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑 ∧ ¬ 𝜓) → ⊥)       (𝜑𝜓)

Theorempm2.21fal 1503 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 386, (disjunction aka logical inclusive 'or') df-or 385, (implies) wi 4, ¬ (not) wn 3, (logical equivalence) df-bi 197, (nand aka Sheffer stroke) df-nan 1446, and (exclusive or) df-xor 1463.

Theoremtruantru 1504 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∧ ⊤) ↔ ⊤)

Theoremtruanfal 1505 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∧ ⊥) ↔ ⊥)

Theoremfalantru 1506 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∧ ⊤) ↔ ⊥)

Theoremfalanfal 1507 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∧ ⊥) ↔ ⊥)

Theoremtruortru 1508 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ∨ ⊤) ↔ ⊤)

Theoremtruorfal 1509 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∨ ⊥) ↔ ⊤)

Theoremfalortru 1510 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∨ ⊤) ↔ ⊤)

Theoremfalorfal 1511 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ∨ ⊥) ↔ ⊥)

Theoremtruimtru 1512 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ → ⊤) ↔ ⊤)

Theoremtruimfal 1513 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ → ⊥) ↔ ⊥)

Theoremfalimtru 1514 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ → ⊤) ↔ ⊤)

Theoremfalimfal 1515 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ → ⊥) ↔ ⊤)

Theoremnottru 1516 A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(¬ ⊤ ↔ ⊥)

Theoremnotfal 1517 A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(¬ ⊥ ↔ ⊤)

Theoremtrubitru 1518 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ↔ ⊤) ↔ ⊤)

Theoremfalbitru 1519 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 1520 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 1521 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ↔ ⊥) ↔ ⊤)

Theoremtrunantru 1522 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ⊼ ⊤) ↔ ⊥)

Theoremtrunanfal 1523 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 1524 A identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ⊼ ⊤) ↔ ⊤)

Theoremfalnanfal 1525 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊥ ⊼ ⊥) ↔ ⊤)

Theoremtruxortru 1526 A identity. (Contributed by David A. Wheeler, 8-May-2015.)
((⊤ ⊻ ⊤) ↔ ⊥)

Theoremtruxorfal 1527 A identity. (Contributed by David A. Wheeler, 8-May-2015.)
((⊤ ⊻ ⊥) ↔ ⊤)

Theoremfalxortru 1528 A identity. (Contributed by David A. Wheeler, 9-May-2015.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
((⊥ ⊻ ⊤) ↔ ⊤)

Theoremfalxorfal 1529 A identity. (Contributed by David A. Wheeler, 9-May-2015.)
((⊥ ⊻ ⊥) ↔ ⊥)

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 1463) and the logical conjunction (df-an 386).

The full adder takes into account an "input carry", so it has three inputs and again two outputs, corresponding to the "sum" (df-had 1531) and "updated carry" (df-cad 1544). 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.

Syntaxwhad 1530 Syntax for the "sum" output of the full adder. (Contributed by Mario Carneiro, 4-Sep-2016.)

Definitiondf-had 1531 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 1532 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 1533 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 1534 Associative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
(hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓𝜒)))

Theoremhadbi 1535 The adder sum is the same as the triple biconditional. (Contributed by Mario Carneiro, 4-Sep-2016.)
(hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ↔ 𝜒))

Theoremhadcoma 1536 Commutative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadcomb 1537 Commutative law for the adders sum. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadrot 1538 Rotation law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremhadnot 1539 The adder sum distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)

Theoremhad1 1540 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 1541 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 1542 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)))

Syntaxwcad 1543 Syntax for the "carry" output of the full adder. (Contributed by Mario Carneiro, 4-Sep-2016.)

Definitiondf-cad 1544 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 1545 and cadan 1546 for alternate definitions. (Contributed by Mario Carneiro, 4-Sep-2016.)
(cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (𝜒 ∧ (𝜑𝜓))))

Theoremcador 1545 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 1546 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 1547 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 1548 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 1549 Commutative law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcadcomb 1550 Commutative law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)

Theoremcadrot 1551 Rotation law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcadnot 1552 The adder carry distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)

Theoremcad1 1553 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 1554 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 1555 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 1556 If (at least) two inputs are true, then the adder carry is true. (Contributed by Mario Carneiro, 4-Sep-2016.)

Theoremcadtru 1557 The adder carry is true as soon as its first two inputs are the truth constant. (Contributed by Mario Carneiro, 4-Sep-2016.)

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 1558. This section proves minimp 1558 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 1558 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))))

Theoremminimp-sylsimp 1559 Derivation of sylsimp (jarr 106) from ax-mp 5 and minimp 1558. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜓𝜒))

Theoremminimp-ax1 1560 Derivation of ax-1 6 from ax-mp 5 and minimp 1558. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))

Theoremminimp-ax2c 1561 Derivation of a commuted form of ax-2 7 from ax-mp 5 and minimp 1558. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))

Theoremminimp-ax2 1562 Derivation of ax-2 7 from ax-mp 5 and minimp 1558. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))

Theoremminimp-pm2.43 1563 Derivation of pm2.43 56 (also called "hilbert" or W) from ax-mp 5 and minimp 1558. It uses the classical derivation from ax-1 6 and ax-2 7 written DD22D21 in D-notation (see head comment for an explanation) and shortens the proof using mp2 9 (which only requires ax-mp 5). (Contributed by BJ, 31-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜑𝜓)) → (𝜑𝜓))

1.3.2  Derive the Lukasiewicz axioms from Meredith's sole axiom

Theoremmeredith 1564 Carew Meredith's sole axiom for propositional calculus. This amazing formula is thought to be the shortest possible single axiom for propositional calculus with inference rule ax-mp 5, where negation and implication are primitive. Here we prove Meredith's axiom from ax-1 6, ax-2 7, and ax-3 8. Then from it we derive the Lukasiewicz axioms luk-1 1578, luk-2 1579, and luk-3 1580. Using these we finally rederive our axioms as ax1 1589, ax2 1590, and ax3 1591, thus proving the equivalence of all three systems. C. A. Meredith, "Single Axioms for the Systems (C,N), (C,O) and (A,N) of the Two-Valued Propositional Calculus," The Journal of Computing Systems vol. 1 (1953), pp. 155-164. Meredith claimed to be close to a proof that this axiom is the shortest possible, but the proof was apparently never completed.

An obscure Irish lecturer, Meredith (1904-1976) became enamored with logic somewhat late in life after attending talks by Lukasiewicz and produced many remarkable results such as this axiom. From his obituary: "He did logic whenever time and opportunity presented themselves, and he did it on whatever materials came to hand: in a pub, his favored pint of porter within reach, he would use the inside of cigarette packs to write proofs for logical colleagues." (Contributed by NM, 14-Dec-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by Wolf Lammen, 28-May-2013.)

(((((𝜑𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒) → 𝜏) → ((𝜏𝜑) → (𝜃𝜑)))

Theoremmerlem1 1565 Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (The step numbers refer to Meredith's original paper.) (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜒 → (¬ 𝜑𝜓)) → 𝜏) → (𝜑𝜏))

Theoremmerlem2 1566 Step 4 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜑) → 𝜒) → (𝜃𝜒))

Theoremmerlem3 1567 Step 7 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜓𝜒) → 𝜑) → (𝜒𝜑))

Theoremmerlem4 1568 Step 8 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜏 → ((𝜏𝜑) → (𝜃𝜑)))

Theoremmerlem5 1569 Step 11 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (¬ ¬ 𝜑𝜓))

Theoremmerlem6 1570 Step 12 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜒 → (((𝜓𝜒) → 𝜑) → (𝜃𝜑)))

Theoremmerlem7 1571 Between steps 14 and 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (((𝜓𝜒) → 𝜃) → (((𝜒𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃)))

Theoremmerlem8 1572 Step 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜓𝜒) → 𝜃) → (((𝜒𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃))

Theoremmerlem9 1573 Step 18 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → (𝜒 → (𝜃 → (𝜓𝜏)))) → (𝜂 → (𝜒 → (𝜃 → (𝜓𝜏)))))

Theoremmerlem10 1574 Step 19 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜑𝜓)) → (𝜃 → (𝜑𝜓)))

Theoremmerlem11 1575 Step 20 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜑𝜓)) → (𝜑𝜓))

Theoremmerlem12 1576 Step 28 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜃 → (¬ ¬ 𝜒𝜒)) → 𝜑) → 𝜑)

Theoremmerlem13 1577 Step 35 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (((𝜃 → (¬ ¬ 𝜒𝜒)) → ¬ ¬ 𝜑) → 𝜓))

Theoremluk-1 1578 1 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Theoremluk-2 1579 2 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑𝜑) → 𝜑)

Theoremluk-3 1580 3 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (¬ 𝜑𝜓))

1.3.3  Derive the standard axioms from the Lukasiewicz axioms

Theoremluklem1 1581 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 23-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)

Theoremluklem2 1582 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → ¬ 𝜓) → (((𝜑𝜒) → 𝜃) → (𝜓𝜃)))

Theoremluklem3 1583 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (((¬ 𝜑𝜓) → 𝜒) → (𝜃𝜒)))

Theoremluklem4 1584 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((((¬ 𝜑𝜑) → 𝜑) → 𝜓) → 𝜓)

Theoremluklem5 1585 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))

Theoremluklem6 1586 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜑𝜓)) → (𝜑𝜓))

Theoremluklem7 1587 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))

Theoremluklem8 1588 Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))

Theoremax1 1589 Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))

Theoremax2 1590 Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))

Theoremax3 1591 Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑))

1.3.4  Derive Nicod's axiom from the standard axioms

Prove Nicod's axiom and implication and negation definitions.

Theoremnic-dfim 1592 Define implication in terms of 'nand'. Analogous to ((𝜑 ⊼ (𝜓𝜓)) ↔ (𝜑𝜓)). In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to a definition (\$a statement). (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑𝜓)) ⊼ (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑 ⊼ (𝜓𝜓))) ⊼ ((𝜑𝜓) ⊼ (𝜑𝜓))))

Theoremnic-dfneg 1593 Define negation in terms of 'nand'. Analogous to ((𝜑𝜑) ↔ ¬ 𝜑). In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to a definition (\$a statement). (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜑) ⊼ ¬ 𝜑) ⊼ (((𝜑𝜑) ⊼ (𝜑𝜑)) ⊼ (¬ 𝜑 ⊼ ¬ 𝜑)))

Theoremnic-mp 1594 Derive Nicod's rule of modus ponens using 'nand', from the standard one. Although the major and minor premise together also imply 𝜒, this form is necessary for useful derivations from nic-ax 1596. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to an axiom (\$a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑 ⊼ (𝜒𝜓))       𝜓

Theoremnic-mpALT 1595 A direct proof of nic-mp 1594. (Contributed by NM, 30-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑 ⊼ (𝜒𝜓))       𝜓

Theoremnic-ax 1596 Nicod's axiom derived from the standard ones. See Introduction to Mathematical Philosophy by B. Russell, p. 152. Like meredith 1564, the usual axioms can be derived from this and vice versa. Unlike meredith 1564, Nicod uses a different connective ('nand'), so another form of modus ponens must be used in proofs, e.g. { nic-ax 1596, nic-mp 1594 } is equivalent to { luk-1 1578, luk-2 1579, luk-3 1580, ax-mp 5 }. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to an axiom (\$a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))

Theoremnic-axALT 1597 A direct proof of nic-ax 1596. (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))

1.3.5  Derive the Lukasiewicz axioms from Nicod's axiom

Theoremnic-imp 1598 Inference for nic-mp 1594 using nic-ax 1596 as major premise. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ⊼ (𝜒𝜓))       ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))

Theoremnic-idlem1 1599 Lemma for nic-id 1601. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜃 ⊼ (𝜏 ⊼ (𝜏𝜏))) ⊼ (((𝜑 ⊼ (𝜒𝜓)) ⊼ 𝜃) ⊼ ((𝜑 ⊼ (𝜒𝜓)) ⊼ 𝜃)))

Theoremnic-idlem2 1600 Lemma for nic-id 1601. Inference used by nic-id 1601. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜂 ⊼ ((𝜑 ⊼ (𝜒𝜓)) ⊼ 𝜃))       ((𝜃 ⊼ (𝜏 ⊼ (𝜏𝜏))) ⊼ 𝜂)

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