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Theorem List for Metamath Proof Explorer - 1601-1700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcadan 1601 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 1602 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 1603 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 1604 Commutative law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
(cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜓, 𝜑, 𝜒))
 
Theoremcadcomb 1605 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 1606 Rotation law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
(cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜓, 𝜒, 𝜑))
 
Theoremcadnot 1607 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 1608 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 1609 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 1610 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 1611 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 1612 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 1613. This section proves minimp 1613 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 7 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 1613 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. Among single axioms of this length, it is the one with simplest antecedents (i.e., in the corresponding ordering of binary trees which first compares left subtrees, it is the first one). (Contributed by BJ, 4-Apr-2021.)
(𝜑 → ((𝜓𝜒) → (((𝜃𝜓) → (𝜒𝜏)) → (𝜓𝜏))))
 
Theoremminimp-syllsimp 1614 Derivation of Syll-Simp (jarr 106) from ax-mp 5 and minimp 1613. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜓𝜒))
 
Theoremminimp-ax1 1615 Derivation of ax-1 6 from ax-mp 5 and minimp 1613. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))
 
Theoremminimp-ax2c 1616 Derivation of a commuted form of ax-2 7 from ax-mp 5 and minimp 1613. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))
 
Theoremminimp-ax2 1617 Derivation of ax-2 7 from ax-mp 5 and minimp 1613. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
 
Theoremminimp-pm2.43 1618 Derivation of pm2.43 56 (also called "hilbert" or W) from ax-mp 5 and minimp 1613. 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  Implicational Calculus

Implicational calculus is the fragment of propositional logic that uses only material implication, and not negation. It can be axiomatized by inference rule modus ponens ax-mp 5 together with the axioms { ax-1 6, ax-2 7, peirce 203 } or the Tarski-Bernays axioms { ax-1 6, imim1 83, peirce 203 } or with the single axiom { impsingle 1619 }. From either one of these three axiom sets, all tautologies containing only material implication may be proved. In contrast, minimal implicational calculus, which is presented just before this section, cannot prove certain tautologies (peirce 203, for example ). The class of theorems proved by minimal implicational calculus is thus a subset of the class of theorems proved by implicational calculus.

The primary source for this section is the paper by Lukasiewicz, The Shortest Axiom of the Implicational Calculus of Propositions, Proceedings of the Royal Irish Academy, Section A, vol. 52 (1948-1950), 25--33. It will be cited as simply "Lukasiewicz" in the remainder of this section.

This section proves that the above three distinct axiom sets for implicational calculus all produce the same class of theorems.

(Contributed by Larry Lesyna and Jeffrey P. Machado, 1-Aug-2023.)

 
Theoremimpsingle 1619 The shortest single axiom for implicational calculus, due to Lukasiewicz. It is now known to be the unique shortest axiom. The axiom is proved here starting from { ax-1 6, ax-2 7 and peirce 203 }. (Contributed by Larry Lesyna and Jeffrey P. Machado, 18-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → ((𝜒𝜑) → (𝜃𝜑)))
 
Theoremimpsingle-step4 1620 Derivation of impsingle-step4 from ax-mp 5 and impsingle 1619. It is used as a lemma in proofs of imim1 83 and peirce 203 from impsingle 1619. It is Step 4 in Lukasiewicz, where it appears as 'CCCpqpCsp' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜑) → (𝜒𝜑))
 
Theoremimpsingle-step8 1621 Derivation of impsingle-step8 from ax-mp 5 and impsingle 1619. It is used as a lemma in proofs of ax-1 6 imim1 83 and peirce 203 from impsingle 1619. It is Step 8 in Lukasiewicz, where it appears as 'CCCsqpCqp' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜓𝜒))
 
Theoremimpsingle-ax1 1622 Derivation of impsingle-ax1 (ax-1 6) from ax-mp 5 and impsingle 1619. Lukasiewicz was used to construct this proof. Every formula corresponding to a detachment step was converted to its corresponding Metamath formula. mmj2 was used to unify each formula using ax-mp 5, which in turn produced this proof. With extremely high confidence, this result shows that the Lukasiewicz proof of ax-1 6 (step 27) is correct and that Metamath correctly verifies the proof. The same comments apply to the proofs that follow this one. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))
 
Theoremimpsingle-step15 1623 Derivation of impsingle-step15 from ax-mp 5 and impsingle 1619. It is used as a lemma in proofs of imim1 83 and peirce 203 from impsingle 1619. It is Step 15 in Lukasiewicz, where it appears as 'CCCrqCspCCrpCsp' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → (𝜒𝜃)) → ((𝜑𝜃) → (𝜒𝜃)))
 
Theoremimpsingle-step18 1624 Derivation of impsingle-step18 from ax-mp 5 and impsingle 1619. It is used as a lemma in proofs of imim1 83 and peirce 203 from impsingle 1619. It is Step 18 in Lukasiewicz, where it appears as 'CCCCrpCspCCCpqrtCuCCCpqrt' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑𝜓) → (𝜒𝜓)) → (((𝜓𝜃) → 𝜑) → 𝜏)) → (𝜂 → (((𝜓𝜃) → 𝜑) → 𝜏)))
 
Theoremimpsingle-step19 1625 Derivation of impsingle-step19 from ax-mp 5 and impsingle 1619. It is used as a lemma in proofs of imim1 83 and peirce 203 from impsingle 1619. It is Step 19 in Lukasiewicz, where it appears as 'CCCCspqCrpCCCpqrCsp' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑𝜓) → 𝜒) → (𝜃𝜓)) → (((𝜓𝜒) → 𝜃) → (𝜑𝜓)))
 
Theoremimpsingle-step20 1626 Derivation of impsingle-step20 from ax-mp 5 and impsingle 1619. It is used as a lemma in proofs of imim1 83 and peirce 203 from impsingle 1619. It is Step 20 in Lukasiewicz, where it appears as 'CCCCrppCspCCCpqrCsp' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑𝜓) → 𝜓) → (𝜒𝜓)) → (((𝜓𝜃) → 𝜑) → (𝜒𝜓)))
 
Theoremimpsingle-step21 1627 Derivation of impsingle-step21 from ax-mp 5 and impsingle 1619. It is used as a lemma in the proof of imim1 83 from impsingle 1619. It is Step 21 in Lukasiewicz, where it appears as 'CCCCprqqCCqrCpr' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑𝜓) → 𝜒) → 𝜒) → ((𝜒𝜓) → (𝜑𝜓)))
 
Theoremimpsingle-step22 1628 Derivation of impsingle-step22 from ax-mp 5 and impsingle 1619. It is used as a lemma in proofs of imim1 83 and peirce 203 from impsingle 1619. It is Step 22 in Lukasiewicz, where it appears as 'Cpp' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜑)
 
Theoremimpsingle-step25 1629 Derivation of impsingle-step25 from ax-mp 5 and impsingle 1619. It is used as a lemma in the proof of imim1 83 from impsingle 1619. It is Step 25 in Lukasiewicz, where it appears as 'CCpqCCCprqq' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (((𝜑𝜒) → 𝜓) → 𝜓))
 
Theoremimpsingle-imim1 1630 Derivation of impsingle-imim1 (imim1 83) from ax-mp 5 and impsingle 1619. It is step 29 in Lukasiewicz. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
 
Theoremimpsingle-peirce 1631 Derivation of impsingle-peirce (peirce 203) from ax-mp 5 and impsingle 1619. It is step 28 in Lukasiewicz. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜑) → 𝜑)
 
Theoremtarski-bernays-ax2 1632 Derivation of ax-2 7 from the Tarski-Bernays axiom set { ax-1 6, imim1 83, peirce 203 }. Note that no inference rule other than ax-mp 5 is used in this proof. This proof establishes a circle of equivalence: From { impsingle 1619 }, { ax-1 6, imim1 83, peirce 203 } was proved. From { ax-1 6, imim1 83, peirce 203 }, { ax-1 6, ax-2 7 and peirce 203 } was proved. From { ax-1 6, ax-2 7 and peirce 203 }, { impsingle 1619 } was proved. Therefore, the theorems that can be proved by selecting any one of these three distinct axiom sets must be equivalent. Prover9 and mmj2 were used to help constuct this proof. (Contributed by Larry Lesyna and Jeffrey P. Machado, 1-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
 
1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom
 
Theoremmeredith 1633 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 1647, luk-2 1648, and luk-3 1649. Using these we finally rederive our axioms as ax1 1658, ax2 1659, and ax3 1660, 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 1634 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 1635 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 1636 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 1637 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 1638 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 1639 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 1640 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 1641 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 1642 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 1643 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 1644 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 1645 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 1646 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 1647 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 1648 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 1649 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))
 
1.3.4  Derive the standard axioms from the Lukasiewicz axioms
 
Theoremluklem1 1650 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 1651 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 1652 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 1653 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 1654 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 1655 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 1656 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 1657 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 1658 Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))
 
Theoremax2 1659 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 1660 Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑))
 
1.3.5  Derive Nicod's axiom from the standard axioms

Prove Nicod's axiom and implication and negation definitions.

 
Theoremnic-dfim 1661 This theorem "defines" implication in terms of 'nand'. Analogous to nanim 1482. 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 1662 This theorem "defines" negation in terms of 'nand'. Analogous to nannot 1483. 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 1663 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 1665. 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 1664 A direct proof of nic-mp 1663. (Contributed by NM, 30-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑 ⊼ (𝜒𝜓))       𝜓
 
Theoremnic-ax 1665 Nicod's axiom derived from the standard ones. See Introduction to Mathematical Philosophy by B. Russell, p. 152. Like meredith 1633, the usual axioms can be derived from this and vice versa. Unlike meredith 1633, Nicod uses a different connective ('nand'), so another form of modus ponens must be used in proofs, e.g. { nic-ax 1665, nic-mp 1663 } is equivalent to { luk-1 1647, luk-2 1648, luk-3 1649, 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 1666 A direct proof of nic-ax 1665. (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))
 
1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom
 
Theoremnic-imp 1667 Inference for nic-mp 1663 using nic-ax 1665 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 1668 Lemma for nic-id 1670. (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 1669 Lemma for nic-id 1670. Inference used by nic-id 1670. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜂 ⊼ ((𝜑 ⊼ (𝜒𝜓)) ⊼ 𝜃))       ((𝜃 ⊼ (𝜏 ⊼ (𝜏𝜏))) ⊼ 𝜂)
 
Theoremnic-id 1670 Theorem id 22 expressed with . (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜏 ⊼ (𝜏𝜏))
 
Theoremnic-swap 1671 The connector is symmetric. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜃𝜑) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))
 
Theoremnic-isw1 1672 Inference version of nic-swap 1671. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃𝜑)       (𝜑𝜃)
 
Theoremnic-isw2 1673 Inference for swapping nested terms. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓 ⊼ (𝜃𝜑))       (𝜓 ⊼ (𝜑𝜃))
 
Theoremnic-iimp1 1674 Inference version of nic-imp 1667 using right-handed term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ⊼ (𝜒𝜓))    &   (𝜃𝜒)       (𝜃𝜑)
 
Theoremnic-iimp2 1675 Inference version of nic-imp 1667 using left-handed term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ⊼ (𝜒𝜒))    &   (𝜃𝜑)       (𝜃 ⊼ (𝜒𝜒))
 
Theoremnic-idel 1676 Inference to remove the trailing term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ⊼ (𝜒𝜓))       (𝜑 ⊼ (𝜒𝜒))
 
Theoremnic-ich 1677 Chained inference. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ⊼ (𝜓𝜓))    &   (𝜓 ⊼ (𝜒𝜒))       (𝜑 ⊼ (𝜒𝜒))
 
Theoremnic-idbl 1678 Double the terms. Since doubling is the same as negation, this can be viewed as a contraposition inference. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ⊼ (𝜓𝜓))       ((𝜓𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜑𝜑)))
 
Theoremnic-bijust 1679 Biconditional justification from Nicod's axiom. For nic-* definitions, the biconditional connective is not used. Instead, definitions are made based on this form. nic-bi1 1680 and nic-bi2 1681 are used to convert the definitions into usable theorems about one side of the implication. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜏𝜏) ⊼ ((𝜏𝜏) ⊼ (𝜏𝜏)))
 
Theoremnic-bi1 1680 Inference to extract one side of an implication from a definition. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜓𝜓)))       (𝜑 ⊼ (𝜓𝜓))
 
Theoremnic-bi2 1681 Inference to extract the other side of an implication from a 'biconditional' definition. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜓𝜓)))       (𝜓 ⊼ (𝜑𝜑))
 
Theoremnic-stdmp 1682 Derive the standard modus ponens from nic-mp 1663. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       𝜓
 
Theoremnic-luk1 1683 Proof of luk-1 1647 from nic-ax 1665 and nic-mp 1663 (and definitions nic-dfim 1661 and nic-dfneg 1662). Note that the standard axioms ax-1 6, ax-2 7, and ax-3 8 are proved from the Lukasiewicz axioms by theorems ax1 1658, ax2 1659, and ax3 1660. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
 
Theoremnic-luk2 1684 Proof of luk-2 1648 from nic-ax 1665 and nic-mp 1663. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑𝜑) → 𝜑)
 
Theoremnic-luk3 1685 Proof of luk-3 1649 from nic-ax 1665 and nic-mp 1663. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (¬ 𝜑𝜓))
 
1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom
 
Theoremlukshef-ax1 1686 This alternative axiom for propositional calculus using the Sheffer Stroke was discovered by Lukasiewicz in his Selected Works. It improves on Nicod's axiom by reducing its number of variables by one.

This axiom also uses nic-mp 1663 for its constructions.

Here, the axiom is proved as a substitution instance of nic-ax 1665. (Contributed by Anthony Hart, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜃 ⊼ (𝜃𝜃)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))
 
Theoremlukshefth1 1687 Lemma for renicax 1689. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜏𝜓) ⊼ ((𝜑𝜏) ⊼ (𝜑𝜏))) ⊼ (𝜃 ⊼ (𝜃𝜃))) ⊼ (𝜑 ⊼ (𝜓𝜒)))
 
Theoremlukshefth2 1688 Lemma for renicax 1689. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜏𝜃) ⊼ ((𝜃𝜏) ⊼ (𝜃𝜏)))
 
Theoremrenicax 1689 A rederivation of nic-ax 1665 from lukshef-ax1 1686, proving that lukshef-ax1 1686 with nic-mp 1663 can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))
 
1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms
 
Theoremtbw-bijust 1690 Justification for tbw-negdf 1691. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ↔ (((𝜑𝜓) → ((𝜓𝜑) → ⊥)) → ⊥))
 
Theoremtbw-negdf 1691 The definition of negation, in terms of and . (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥)
 
Theoremtbw-ax1 1692 The first of four axioms in the Tarski-Bernays-Wajsberg system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
 
Theoremtbw-ax2 1693 The second of four axioms in the Tarski-Bernays-Wajsberg system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))
 
Theoremtbw-ax3 1694 The third of four axioms in the Tarski-Bernays-Wajsberg system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜑) → 𝜑)
 
Theoremtbw-ax4 1695 The fourth of four axioms in the Tarski-Bernays-Wajsberg system.

This axiom was added to the Tarski-Bernays axiom system (see tb-ax1 33629, tb-ax2 33630, and tb-ax3 33631) by Wajsberg for completeness. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

(⊥ → 𝜑)
 
Theoremtbwsyl 1696 Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)
 
Theoremtbwlem1 1697 Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))
 
Theoremtbwlem2 1698 Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓 → ⊥)) → (((𝜑𝜒) → 𝜃) → (𝜓𝜃)))
 
Theoremtbwlem3 1699 Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((((𝜑 → ⊥) → 𝜑) → 𝜑) → 𝜓) → 𝜓)
 
Theoremtbwlem4 1700 Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑 → ⊥) → 𝜓) → ((𝜓 → ⊥) → 𝜑))
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