P. 17 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	trunanfal 1601	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 1602	A ⊼ identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((⊥ ⊼ ⊤) ↔ ⊤)
Theorem	falnanfal 1603	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 1604	A ⊻ identity. (Contributed by David A. Wheeler, 8-May-2015.)
⊢ ((⊤ ⊻ ⊤) ↔ ⊥)
Theorem	truxorfal 1605	A ⊻ identity. (Contributed by David A. Wheeler, 8-May-2015.)
⊢ ((⊤ ⊻ ⊥) ↔ ⊤)
Theorem	falxortru 1606	A ⊻ identity. (Contributed by David A. Wheeler, 9-May-2015.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
⊢ ((⊥ ⊻ ⊤) ↔ ⊤)
Theorem	falxorfal 1607	A ⊻ identity. (Contributed by David A. Wheeler, 9-May-2015.)
⊢ ((⊥ ⊻ ⊥) ↔ ⊥)
1.2.16.8  Joint denial
Theorem	trunortru 1608	A ⊽ identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 7-Dec-2023.)
⊢ ((⊤ ⊽ ⊤) ↔ ⊥)
Theorem	trunorfal 1609	A ⊽ identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
⊢ ((⊤ ⊽ ⊥) ↔ ⊥)
Theorem	falnortru 1610	A ⊽ identity. (Contributed by Remi, 25-Oct-2023.)
⊢ ((⊥ ⊽ ⊤) ↔ ⊥)
Theorem	falnorfal 1611	A ⊽ identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
⊢ ((⊥ ⊽ ⊥) ↔ ⊤)
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 1531) and the logical conjunction (df-an 400). The full adder takes into account an "input carry", so it has three inputs and again two outputs, corresponding to the "sum" (df-had 1613) and "updated carry" (df-cad 1626). 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 1612	Syntax for the "sum" output of the full adder. (Contributed by Mario Carneiro, 4-Sep-2016.)
wff hadd(𝜑, 𝜓, 𝜒)
Definition	df-had 1613	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 1614	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)))
Theorem	hadbi123i 1615	Equality theorem for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    &   ⊢ (𝜏 ↔ 𝜂)    ⇒   ⊢ (hadd(𝜑, 𝜒, 𝜏) ↔ hadd(𝜓, 𝜃, 𝜂))
Theorem	hadass 1616	Associative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓 ⊻ 𝜒)))
Theorem	hadbi 1617	The adder sum is the same as the triple biconditional. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ 𝜒))
Theorem	hadcoma 1618	Commutative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜑, 𝜒))
Theorem	hadcomb 1619	Commutative law for the adders sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜑, 𝜒, 𝜓))
Theorem	hadrot 1620	Rotation law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜒, 𝜑))
Theorem	hadnot 1621	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))
Theorem	had1 1622	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) ↔ (𝜓 ↔ 𝜒)))
Theorem	had0 1623	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) ↔ (𝜓 ⊻ 𝜒)))
Theorem	hadifp 1624	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, (𝜓 ↔ 𝜒), (𝜓 ⊻ 𝜒)))
1.2.17.2  Full adder: carry
Syntax	wcad 1625	Syntax for the "carry" output of the full adder. (Contributed by Mario Carneiro, 4-Sep-2016.)
wff cadd(𝜑, 𝜓, 𝜒)
Definition	df-cad 1626	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 1627 and cadan 1628 for alternate definitions. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))))
Theorem	cador 1627	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) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒)))
Theorem	cadan 1628	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) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))
Theorem	cadbi123d 1629	Equality theorem for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    &   ⊢ (𝜑 → (𝜂 ↔ 𝜁))    ⇒   ⊢ (𝜑 → (cadd(𝜓, 𝜃, 𝜂) ↔ cadd(𝜒, 𝜏, 𝜁)))
Theorem	cadbi123i 1630	Equality theorem for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    &   ⊢ (𝜏 ↔ 𝜂)    ⇒   ⊢ (cadd(𝜑, 𝜒, 𝜏) ↔ cadd(𝜓, 𝜃, 𝜂))
Theorem	cadcoma 1631	Commutative law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜓, 𝜑, 𝜒))
Theorem	cadcomb 1632	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))
Theorem	cadrot 1633	Rotation law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜓, 𝜒, 𝜑))
Theorem	cadnot 1634	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))
Theorem	cad11 1635	If (at least) two inputs are true, then the adder carry is true. (Contributed by Mario Carneiro, 4-Sep-2016.)
⊢ ((𝜑 ∧ 𝜓) → cadd(𝜑, 𝜓, 𝜒))
Theorem	cad1 1636	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) ↔ (𝜑 ∨ 𝜓)))
Theorem	cad0 1637	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.) (Proof shortened by Wolf Lammen, 21-Sep-2024.)
⊢ (¬ 𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∧ 𝜓)))
Theorem	cadifp 1638	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, (𝜑 ∨ 𝜓), (𝜑 ∧ 𝜓)))
Theorem	cadtru 1639	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 1640. This section proves minimp 1640 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.)
Theorem	minimp 1640	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 → ((𝜓 → 𝜒) → (((𝜃 → 𝜓) → (𝜒 → 𝜏)) → (𝜓 → 𝜏))))
Theorem	minimp-syllsimp 1641	Derivation of Syll-Simp (jarr 106) from ax-mp 5 and minimp 1640. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))
Theorem	minimp-ax1 1642	Derivation of ax-1 6 from ax-mp 5 and minimp 1640. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜑))
Theorem	minimp-ax2c 1643	Derivation of a commuted form of ax-2 7 from ax-mp 5 and minimp 1640. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))
Theorem	minimp-ax2 1644	Derivation of ax-2 7 from ax-mp 5 and minimp 1640. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
Theorem	minimp-pm2.43 1645	Derivation of pm2.43 56 (also called "hilbert" or W) from ax-mp 5 and minimp 1640. 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 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))
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 204 } or the Tarski-Bernays axioms { ax-1 6, imim1 83, peirce 204 } or with the single axiom { impsingle 1646 }. 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 204, 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 Jan 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.)
Theorem	impsingle 1646	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 204 }. (Contributed by Larry Lesyna and Jeffrey P. Machado, 18-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜒 → 𝜑) → (𝜃 → 𝜑)))
Theorem	impsingle-step4 1647	Derivation of impsingle-step4 from ax-mp 5 and impsingle 1646. It is used as a lemma in proofs of imim1 83 and peirce 204 from impsingle 1646. 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.)
⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜒 → 𝜑))
Theorem	impsingle-step8 1648	Derivation of impsingle-step8 from ax-mp 5 and impsingle 1646. It is used as a lemma in proofs of ax-1 6 imim1 83 and peirce 204 from impsingle 1646. 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.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))
Theorem	impsingle-ax1 1649	Derivation of impsingle-ax1 (ax-1 6) from ax-mp 5 and impsingle 1646. 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 → (𝜓 → 𝜑))
Theorem	impsingle-step15 1650	Derivation of impsingle-step15 from ax-mp 5 and impsingle 1646. It is used as a lemma in proofs of imim1 83 and peirce 204 from impsingle 1646. 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.)
⊢ (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃)))
Theorem	impsingle-step18 1651	Derivation of impsingle-step18 from ax-mp 5 and impsingle 1646. It is used as a lemma in proofs of imim1 83 and peirce 204 from impsingle 1646. 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.)
⊢ ((((𝜑 → 𝜓) → (𝜒 → 𝜓)) → (((𝜓 → 𝜃) → 𝜑) → 𝜏)) → (𝜂 → (((𝜓 → 𝜃) → 𝜑) → 𝜏)))
Theorem	impsingle-step19 1652	Derivation of impsingle-step19 from ax-mp 5 and impsingle 1646. It is used as a lemma in proofs of imim1 83 and peirce 204 from impsingle 1646. 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.)
⊢ ((((𝜑 → 𝜓) → 𝜒) → (𝜃 → 𝜓)) → (((𝜓 → 𝜒) → 𝜃) → (𝜑 → 𝜓)))
Theorem	impsingle-step20 1653	Derivation of impsingle-step20 from ax-mp 5 and impsingle 1646. It is used as a lemma in proofs of imim1 83 and peirce 204 from impsingle 1646. 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.)
⊢ ((((𝜑 → 𝜓) → 𝜓) → (𝜒 → 𝜓)) → (((𝜓 → 𝜃) → 𝜑) → (𝜒 → 𝜓)))
Theorem	impsingle-step21 1654	Derivation of impsingle-step21 from ax-mp 5 and impsingle 1646. It is used as a lemma in the proof of imim1 83 from impsingle 1646. 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.)
⊢ ((((𝜑 → 𝜓) → 𝜒) → 𝜒) → ((𝜒 → 𝜓) → (𝜑 → 𝜓)))
Theorem	impsingle-step22 1655	Derivation of impsingle-step22 from ax-mp 5 and impsingle 1646. It is used as a lemma in proofs of imim1 83 and peirce 204 from impsingle 1646. 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.)
⊢ (𝜑 → 𝜑)
Theorem	impsingle-step25 1656	Derivation of impsingle-step25 from ax-mp 5 and impsingle 1646. It is used as a lemma in the proof of imim1 83 from impsingle 1646. 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.)
⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → 𝜓) → 𝜓))
Theorem	impsingle-imim1 1657	Derivation of impsingle-imim1 (imim1 83) from ax-mp 5 and impsingle 1646. 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.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
Theorem	impsingle-peirce 1658	Derivation of impsingle-peirce (peirce 204) from ax-mp 5 and impsingle 1646. 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.)
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
Theorem	tarski-bernays-ax2 1659	Derivation of ax-2 7 from the Tarski-Bernays axiom set { ax-1 6, imim1 83, peirce 204 }. Note that no inference rule other than ax-mp 5 is used in this proof. This proof establishes a circle of equivalence: From { impsingle 1646 }, { ax-1 6, imim1 83, peirce 204 } was proved. From { ax-1 6, imim1 83, peirce 204 }, { ax-1 6, ax-2 7 and peirce 204 } was proved. From { ax-1 6, ax-2 7 and peirce 204 }, { impsingle 1646 } 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 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom
Theorem	meredith 1660	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 1674, luk-2 1675, and luk-3 1676. Using these we finally rederive our axioms as ax1 1685, ax2 1686, and ax3 1687, 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.)
⊢ (((((𝜑 → 𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒) → 𝜏) → ((𝜏 → 𝜑) → (𝜃 → 𝜑)))
Theorem	merlem1 1661	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 → (¬ 𝜑 → 𝜓)) → 𝜏) → (𝜑 → 𝜏))
Theorem	merlem2 1662	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.)
⊢ (((𝜑 → 𝜑) → 𝜒) → (𝜃 → 𝜒))
Theorem	merlem3 1663	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.)
⊢ (((𝜓 → 𝜒) → 𝜑) → (𝜒 → 𝜑))
Theorem	merlem4 1664	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 → ((𝜏 → 𝜑) → (𝜃 → 𝜑)))
Theorem	merlem5 1665	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.)
⊢ ((𝜑 → 𝜓) → (¬ ¬ 𝜑 → 𝜓))
Theorem	merlem6 1666	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 → (((𝜓 → 𝜒) → 𝜑) → (𝜃 → 𝜑)))
Theorem	merlem7 1667	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 → (((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃)))
Theorem	merlem8 1668	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.)
⊢ (((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃))
Theorem	merlem9 1669	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.)
⊢ (((𝜑 → 𝜓) → (𝜒 → (𝜃 → (𝜓 → 𝜏)))) → (𝜂 → (𝜒 → (𝜃 → (𝜓 → 𝜏)))))
Theorem	merlem10 1670	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 → (𝜑 → 𝜓)) → (𝜃 → (𝜑 → 𝜓)))
Theorem	merlem11 1671	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 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))
Theorem	merlem12 1672	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 → (¬ ¬ 𝜒 → 𝜒)) → 𝜑) → 𝜑)
Theorem	merlem13 1673	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.)
⊢ ((𝜑 → 𝜓) → (((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑) → 𝜓))
Theorem	luk-1 1674	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.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
Theorem	luk-2 1675	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.)
⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
Theorem	luk-3 1676	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 → (¬ 𝜑 → 𝜓))
1.3.4  Derive the standard axioms from the Lukasiewicz axioms
Theorem	luklem1 1677	Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 23-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	luklem2 1678	Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → ¬ 𝜓) → (((𝜑 → 𝜒) → 𝜃) → (𝜓 → 𝜃)))
Theorem	luklem3 1679	Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (((¬ 𝜑 → 𝜓) → 𝜒) → (𝜃 → 𝜒)))
Theorem	luklem4 1680	Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((¬ 𝜑 → 𝜑) → 𝜑) → 𝜓) → 𝜓)
Theorem	luklem5 1681	Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜑))
Theorem	luklem6 1682	Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))
Theorem	luklem7 1683	Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
Theorem	luklem8 1684	Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))
Theorem	ax1 1685	Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜑))
Theorem	ax2 1686	Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
Theorem	ax3 1687	Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))
1.3.5  Derive Nicod's axiom from the standard axioms Prove Nicod's axiom and implication and negation definitions.
Theorem	nic-dfim 1688	This theorem "defines" implication in terms of 'nand'. Analogous to nanim 1517. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to a definition ($a statement). (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 → 𝜓)) ⊼ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 ⊼ (𝜓 ⊼ 𝜓))) ⊼ ((𝜑 → 𝜓) ⊼ (𝜑 → 𝜓))))
Theorem	nic-dfneg 1689	This theorem "defines" negation in terms of 'nand'. Analogous to nannot 1518. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to a definition ($a statement). (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 ⊼ 𝜑) ⊼ ¬ 𝜑) ⊼ (((𝜑 ⊼ 𝜑) ⊼ (𝜑 ⊼ 𝜑)) ⊼ (¬ 𝜑 ⊼ ¬ 𝜑)))
Theorem	nic-mp 1690	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 1692. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to an axiom ($a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓))    ⇒   ⊢ 𝜓
Theorem	nic-mpALT 1691	A direct proof of nic-mp 1690. (Contributed by NM, 30-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓))    ⇒   ⊢ 𝜓
Theorem	nic-ax 1692	Nicod's axiom derived from the standard ones. See Introduction to Mathematical Philosophy by B. Russell, p. 152. Like meredith 1660, the usual axioms can be derived from this and vice versa. Unlike meredith 1660, Nicod uses a different connective ('nand'), so another form of modus ponens must be used in proofs, e.g., { nic-ax 1692, nic-mp 1690 } is equivalent to { luk-1 1674, luk-2 1675, luk-3 1676, ax-mp 5 }. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to an axiom ($a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))
Theorem	nic-axALT 1693	A direct proof of nic-ax 1692. (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))
1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom
Theorem	nic-imp 1694	Inference for nic-mp 1690 using nic-ax 1692 as major premise. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓))    ⇒   ⊢ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))
Theorem	nic-idlem1 1695	Lemma for nic-id 1697. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜃 ⊼ (𝜏 ⊼ (𝜏 ⊼ 𝜏))) ⊼ (((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ 𝜃) ⊼ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ 𝜃)))
Theorem	nic-idlem2 1696	Lemma for nic-id 1697. Inference used by nic-id 1697. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜂 ⊼ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ 𝜃))    ⇒   ⊢ ((𝜃 ⊼ (𝜏 ⊼ (𝜏 ⊼ 𝜏))) ⊼ 𝜂)
Theorem	nic-id 1697	Theorem id 22 expressed with ⊼. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜏 ⊼ (𝜏 ⊼ 𝜏))
Theorem	nic-swap 1698	The connector ⊼ is symmetric. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜃 ⊼ 𝜑) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))
Theorem	nic-isw1 1699	Inference version of nic-swap 1698. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜃 ⊼ 𝜑)    ⇒   ⊢ (𝜑 ⊼ 𝜃)
Theorem	nic-isw2 1700	Inference for swapping nested terms. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜓 ⊼ (𝜃 ⊼ 𝜑))    ⇒   ⊢ (𝜓 ⊼ (𝜑 ⊼ 𝜃))