mmtheorems10 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 901-1000 - Page 10 of 105

Type	Label	Description
Statement
Theorem	mp3an3 901	An inference based on modus ponens.
|- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	mp3an12 902	An inference based on modus ponens.
|- ph   &   |- ps   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ch -> th)
Theorem	mp3an13 903	An inference based on modus ponens.
|- ph   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ps -> th)
Theorem	mp3an23 904	An inference based on modus ponens.
|- ps   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> th)
Theorem	mp3an1i 905	An inference based on modus ponens.
|- ps   &   |- (ph -> ((ps /\ ch /\ th) -> ta))   =>   |- (ph -> ((ch /\ th) -> ta))
Theorem	mp3anl1 906	An inference based on modus ponens.
|- ph   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ps /\ ch) /\ th) -> ta)
Theorem	mp3anl2 907	An inference based on modus ponens.
|- ps   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ph /\ ch) /\ th) -> ta)
Theorem	mp3anl3 908	An inference based on modus ponens.
|- ch   &   |- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ph /\ ps) /\ th) -> ta)
Theorem	mp3anr1 909	An inference based on modus ponens.
|- ps   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ch /\ th)) -> ta)
Theorem	mp3anr2 910	An inference based on modus ponens.
|- ch   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ps /\ th)) -> ta)
Theorem	mp3anr3 911	An inference based on modus ponens.
|- th   &   |- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- ((ph /\ (ps /\ ch)) -> ta)
Theorem	mp3an 912	An inference based on modus ponens.
|- ph   &   |- ps   &   |- ch   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- th
Theorem	mpd3an3 913	An inference based on modus ponens.
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	mpd3an23 914	An inference based on modus ponens.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> th)
Theorem	biimp3a 915	Infer implication from a logical equivalence. Similar to biimpa 416.
|- ((ph /\ ps) -> (ch <-> th))   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3anandis 916	Inference that undistributes a triple conjunction in the antecedent.
|- (((ph /\ ps) /\ (ph /\ ch) /\ (ph /\ th)) -> ta)   =>   |- ((ph /\ (ps /\ ch /\ th)) -> ta)
Theorem	3anandirs 917	Inference that undistributes a triple conjunction in the antecedent.
|- (((ph /\ th) /\ (ps /\ th) /\ (ch /\ th)) -> ta)   =>   |- (((ph /\ ps /\ ch) /\ th) -> ta)
Theorem	ecase23d 918	Deduction for elimination by cases.
|- (ph -> -. ch)   &   |- (ph -> -. th)   &   |- (ph -> (ps \/ ch \/ th))   =>   |- (ph -> ps)
Theorem	3ecase 919	Inference for elimination by cases.
|- (-. ph -> th)   &   |- (-. ps -> th)   &   |- (-. ch -> th)   &   |- ((ph /\ ps /\ ch) -> th)   =>   |- th
Other axiomatizations of classical propositional calculus
Theorem	meredith 920	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 7, where negation and implication are primitive. Here we prove Meredith's axiom from ax-1 4, ax-2 5, and ax-3 6. Then from it we derive the Lukasiewicz axioms luk-1 934, luk-2 935, and luk-3 936. Using these we finally re-derive our axioms as ax1 945, ax2 946, and ax3 947, 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. 3 (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."
|- (((((ph -> ps) -> (-. ch -> -. th)) -> ch) -> ta) -> ((ta -> ph) -> (th -> ph)))
Theorem	merlem1 921	Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (The step numbers refer to Meredith's original paper.)
|- (((ch -> (-. ph -> ps)) -> ta) -> (ph -> ta))
Theorem	merlem2 922	Step 4 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((ph -> ph) -> ch) -> (th -> ch))
Theorem	merlem3 923	Step 7 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((ps -> ch) -> ph) -> (ch -> ph))
Theorem	merlem4 924	Step 8 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (ta -> ((ta -> ph) -> (th -> ph)))
Theorem	merlem5 925	Step 11 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- ((ph -> ps) -> (-. -. ph -> ps))
Theorem	merlem6 926	Step 12 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (ch -> (((ps -> ch) -> ph) -> (th -> ph)))
Theorem	merlem7 927	Between steps 14 and 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (ph -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)))
Theorem	merlem8 928	Step 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th))
Theorem	merlem9 929	Step 18 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((ph -> ps) -> (ch -> (th -> (ps -> ta)))) -> (et -> (ch -> (th -> (ps -> ta)))))
Theorem	merlem10 930	Step 19 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- ((ph -> (ph -> ps)) -> (th -> (ph -> ps)))
Theorem	merlem11 931	Step 20 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- ((ph -> (ph -> ps)) -> (ph -> ps))
Theorem	merlem12 932	Step 28 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- (((th -> (-. -. ch -> ch)) -> ph) -> ph)
Theorem	merlem13 933	Step 35 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
|- ((ph -> ps) -> (((th -> (-. -. ch -> ch)) -> -. -. ph) -> ps))
Theorem	luk-1 934	1 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom.
|- ((ph -> ps) -> ((ps -> ch) -> (ph -> ch)))
Theorem	luk-2 935	2 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom.
|- ((-. ph -> ph) -> ph)
Theorem	luk-3 936	3 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom.
|- (ph -> (-. ph -> ps))
Theorem	luklem1 937	Used to rederive standard propositional axioms from Lukasiewicz'.
|- (ph -> ps)   &   |- (ps -> ch)   =>   |- (ph -> ch)
Theorem	luklem2 938	Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((ph -> -. ps) -> (((ph -> ch) -> th) -> (ps -> th)))
Theorem	luklem3 939	Used to rederive standard propositional axioms from Lukasiewicz'.
|- (ph -> (((-. ph -> ps) -> ch) -> (th -> ch)))
Theorem	luklem4 940	Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((((-. ph -> ph) -> ph) -> ps) -> ps)
Theorem	luklem5 941	Used to rederive standard propositional axioms from Lukasiewicz'.
|- (ph -> (ps -> ph))
Theorem	luklem6 942	Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((ph -> (ph -> ps)) -> (ph -> ps))
Theorem	luklem7 943	Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((ph -> (ps -> ch)) -> (ps -> (ph -> ch)))
Theorem	luklem8 944	Used to rederive standard propositional axioms from Lukasiewicz'.
|- ((ph -> ps) -> ((ch -> ph) -> (ch -> ps)))
Theorem	ax1 945	Standard propositional axiom derived from Lukasiewicz axioms.
|- (ph -> (ps -> ph))
Theorem	ax2 946	Standard propositional axiom derived from Lukasiewicz axioms.
|- ((ph -> (ps -> ch)) -> ((ph -> ps) -> (ph -> ch)))
Theorem	ax3 947	Standard propositional axiom derived from Lukasiewicz axioms.
|- ((-. ph -> -. ps) -> (ps -> ph))
Theorem	nicodraw 948	Axiom of Nicod from Introduction to Mathematical Philosophy B. Russell, p. 152. The axiom is recovered from this raw form by substituting (ph | ps) for -. (ph /\ ps), where | is the Sheffer stroke (NAND) connective, so that the Sheffer stroke becomes the sole connective. See nicodmpraw 949 for the inference rule. (Based on a proof by Jeff Hoffman, 19-Nov-2007.)
|- -. (-. (ph /\ -. (ch /\ ps)) /\ -. (-. (ta /\ -. (ta /\ ta)) /\ -. (-. (th /\ ch) /\ -. (-. (ph /\ th) /\ -. (ph /\ th)))))
Theorem	nicodmpraw 949	The inference rule for the axiom of Nicod, in raw form as explained in nicodraw 948.
|- ph   &   |- -. (ph /\ -. (ch /\ ps))   =>   |- ps
Predicate calculus without distinct variables
The axioms of "pure" predicate calculus
Syntax	wal 950	Extend wff definition to include the universal quantifier ('for all'). A.xph is read "ph (phi) is true for all x." Typically, in its final application ph 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 A.xph
Axiom	ax-4 951	Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all x, it is true for any specific x (that would typically occur as a free variable in the wff substituted for ph). (A free variable is one that does not occur in the scope of a quantifier: x and y are both free in x = y, but only y is free in A.xx = y.) This is one of the 4 axioms of what we call "pure" predicate calculus. Unlike the more typical textbook Axiom of Specialization, we cannot choose a variable different from x for the special case. That is dealt with later when substitution is introduced - see stdpc4 1168. Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77). Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 955. Conditional forms of the converse are given by ax-12 1104, ax-15 1109, ax-16 1194, and ax-17 1190. General remarks: Predicate calculus, or first-order logic, introduces quantifiers to make statements such as "for all individuals, such-and-such is true" and "there exist individuals such that... ." We introduce a new kind of variable, called an "individual variable," that ranges over individuals. (Actually, in Metamath we are introducing "metavariables" that range over the individual variables of textbook predicate calculus, but the theorems look the same. This is a technical point you should be aware of when studying standard textbooks.) In addition, predicate calculus introduces one or more "predicate symbols" that combine individual variables to form wff"s. We will be concerned with two predicate symbols, the equality sign = used in all of mathematics and the stylized epsilon e. used to express "is an element of" in set theory. Our axioms look quite different from those in standard textbooks, but the rules for manipulating the symbols end up being considerably simpler. The axioms of standard textbooks are derived as theorems stdpc4 1168 and stdpc5 1034. We will work with the axioms for predicate calculus in four phases. Phase 1 introduces "pure" predicate calculus, which has no predicate symbols. Phase 2, starting at ax-8 1101, introduces the predicate symbol for equality. Phase 3, starting at ax-13 1107, introduces the stylized epsilon predicate symbol for set theory (without specifying any of its properties that are peculiar to set theory). Phase 4, starting at ax-17 1190, introduces the concept of distinct variables (our first use of the $d statement). In phase 3, we will define (df-sb 1155) and develop the concept of proper substitution. In standard textbooks, substitution is introduced immediately with a somewhat complex recursive definition, since it is needed to state the axioms. Instead, we will define it in terms of concepts contained in the axioms so that in principle it can be eliminated from the language entirely. Finally, we will define existential uniqueness (df-eu 1359) and develop some basic facts about it. ax-4 951 through ax-7 954 are the axioms of what we call "pure" predicate calculus. These are valid even when their quantified variables x and y occur ("free" or not) in wff's ph and ps. Thus we do not have to worry about "free" variable restrictions that complicate the traditional textbook axioms. An alternate axiomatization could use ax467 997 in place of ax-4 951, ax-6 953, and ax-7 954.
|- (A.xph -> ph)
Axiom	ax-5 952	Axiom of Quantified Implication. This axiom moves a quantifier from outside to inside an implication, quantifying ps. Notice that x must not be a free variable in the antecedent of the quantified implication, and we express this by binding ph to "protect" the axiom from a ph containing a free x. One of the 4 axioms of pure predicate calculus. Axiom scheme C4' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Lemma 5 of [Monk2] p. 108 and Axiom 5 of [Mendelson] p. 69.
|- (A.x(A.xph -> ps) -> (A.xph -> A.xps))
Axiom	ax-6 953	Axiom of Quantified Negation. This axiom is used to manipulate negated quantifiers. One of the 4 axioms of pure predicate calculus. Equivalent to axiom scheme C7' in [Megill] p. 448 (p. 16 of the preprint). Another equivalent variant ax6-2 1001 appears as Axiom C5-2 of [Monk2] p. 113. An alternate axiomatization could use ax467 997 in place of ax-4 951, ax-6 953, and ax-7 954.
|- (-. A.x -. A.xph -> ph)
Axiom	ax-7 954	Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. One of the 4 axioms of pure predicate calculus. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109. An alternate axiomatization could use ax467 997 in place of ax-4 951, ax-6 953, and ax-7 954.