mmtheorems10 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 901-1000 - Page 10 of 107

Type	Label	Description
Statement
Theorem	an6 901	Rearrangement of 6 conjuncts.
⊢ (((φ ⋀ ψ ⋀ χ) ⋀ (θ ⋀ τ ⋀ η)) ↔ ((φ ⋀ θ) ⋀ (ψ ⋀ τ) ⋀ (χ ⋀ η)))
Theorem	mp3an1 902	An inference based on modus ponens.
⊢ φ    &   ⊢ ((φ ⋀ ψ ⋀ χ) → θ)    ⇒   ⊢ ((ψ ⋀ χ) → θ)
Theorem	mp3an2 903	An inference based on modus ponens.
⊢ ψ    &   ⊢ ((φ ⋀ ψ ⋀ χ) → θ)    ⇒   ⊢ ((φ ⋀ χ) → θ)
Theorem	mp3an3 904	An inference based on modus ponens.
⊢ χ    &   ⊢ ((φ ⋀ ψ ⋀ χ) → θ)    ⇒   ⊢ ((φ ⋀ ψ) → θ)
Theorem	mp3an12 905	An inference based on modus ponens.
⊢ φ    &   ⊢ ψ    &   ⊢ ((φ ⋀ ψ ⋀ χ) → θ)    ⇒   ⊢ (χ → θ)
Theorem	mp3an13 906	An inference based on modus ponens.
⊢ φ    &   ⊢ χ    &   ⊢ ((φ ⋀ ψ ⋀ χ) → θ)    ⇒   ⊢ (ψ → θ)
Theorem	mp3an23 907	An inference based on modus ponens.
⊢ ψ    &   ⊢ χ    &   ⊢ ((φ ⋀ ψ ⋀ χ) → θ)    ⇒   ⊢ (φ → θ)
Theorem	mp3an1i 908	An inference based on modus ponens.
⊢ ψ    &   ⊢ (φ → ((ψ ⋀ χ ⋀ θ) → τ))    ⇒   ⊢ (φ → ((χ ⋀ θ) → τ))
Theorem	mp3anl1 909	An inference based on modus ponens.
⊢ φ    &   ⊢ (((φ ⋀ ψ ⋀ χ) ⋀ θ) → τ)    ⇒   ⊢ (((ψ ⋀ χ) ⋀ θ) → τ)
Theorem	mp3anl2 910	An inference based on modus ponens.
⊢ ψ    &   ⊢ (((φ ⋀ ψ ⋀ χ) ⋀ θ) → τ)    ⇒   ⊢ (((φ ⋀ χ) ⋀ θ) → τ)
Theorem	mp3anl3 911	An inference based on modus ponens.
⊢ χ    &   ⊢ (((φ ⋀ ψ ⋀ χ) ⋀ θ) → τ)    ⇒   ⊢ (((φ ⋀ ψ) ⋀ θ) → τ)
Theorem	mp3anr1 912	An inference based on modus ponens.
⊢ ψ    &   ⊢ ((φ ⋀ (ψ ⋀ χ ⋀ θ)) → τ)    ⇒   ⊢ ((φ ⋀ (χ ⋀ θ)) → τ)
Theorem	mp3anr2 913	An inference based on modus ponens.
⊢ χ    &   ⊢ ((φ ⋀ (ψ ⋀ χ ⋀ θ)) → τ)    ⇒   ⊢ ((φ ⋀ (ψ ⋀ θ)) → τ)
Theorem	mp3anr3 914	An inference based on modus ponens.
⊢ θ    &   ⊢ ((φ ⋀ (ψ ⋀ χ ⋀ θ)) → τ)    ⇒   ⊢ ((φ ⋀ (ψ ⋀ χ)) → τ)
Theorem	mp3an 915	An inference based on modus ponens.
⊢ φ    &   ⊢ ψ    &   ⊢ χ    &   ⊢ ((φ ⋀ ψ ⋀ χ) → θ)    ⇒   ⊢ θ
Theorem	mpd3an3 916	An inference based on modus ponens.
⊢ ((φ ⋀ ψ) → χ)    &   ⊢ ((φ ⋀ ψ ⋀ χ) → θ)    ⇒   ⊢ ((φ ⋀ ψ) → θ)
Theorem	mpd3an23 917	An inference based on modus ponens.
⊢ (φ → ψ)    &   ⊢ (φ → χ)    &   ⊢ ((φ ⋀ ψ ⋀ χ) → θ)    ⇒   ⊢ (φ → θ)
Theorem	biimp3a 918	Infer implication from a logical equivalence. Similar to biimpa 416.
⊢ ((φ ⋀ ψ) → (χ ↔ θ))    ⇒   ⊢ ((φ ⋀ ψ ⋀ χ) → θ)
Theorem	3anandis 919	Inference that undistributes a triple conjunction in the antecedent.
⊢ (((φ ⋀ ψ) ⋀ (φ ⋀ χ) ⋀ (φ ⋀ θ)) → τ)    ⇒   ⊢ ((φ ⋀ (ψ ⋀ χ ⋀ θ)) → τ)
Theorem	3anandirs 920	Inference that undistributes a triple conjunction in the antecedent.
⊢ (((φ ⋀ θ) ⋀ (ψ ⋀ θ) ⋀ (χ ⋀ θ)) → τ)    ⇒   ⊢ (((φ ⋀ ψ ⋀ χ) ⋀ θ) → τ)
Theorem	ecase23d 921	Deduction for elimination by cases.
⊢ (φ → ¬ χ)    &   ⊢ (φ → ¬ θ)    &   ⊢ (φ → (ψ ⋁ χ ⋁ θ))    ⇒   ⊢ (φ → ψ)
Theorem	3ecase 922	Inference for elimination by cases.
⊢ (¬ φ → θ)    &   ⊢ (¬ ψ → θ)    &   ⊢ (¬ χ → θ)    &   ⊢ ((φ ⋀ ψ ⋀ χ) → θ)    ⇒   ⊢ θ
Other axiomatizations of classical propositional calculus
Theorem	meredith 923	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-1 937, luk-2 938, and luk-3 939. Using these we finally re-derive our axioms as ax1 948, ax2 949, and ax3 950, 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."
⊢ (((((φ → ψ) → (¬ χ → ¬ θ)) → χ) → τ) → ((τ → φ) → (θ → φ)))
Theorem	merlem1 924	Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (The step numbers refer to Meredith's original paper.)
⊢ (((χ → (¬ φ → ψ)) → τ) → (φ → τ))
Theorem	merlem2 925	Step 4 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
⊢ (((φ → φ) → χ) → (θ → χ))
Theorem	merlem3 926	Step 7 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
⊢ (((ψ → χ) → φ) → (χ → φ))
Theorem	merlem4 927	Step 8 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
⊢ (τ → ((τ → φ) → (θ → φ)))
Theorem	merlem5 928	Step 11 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
⊢ ((φ → ψ) → (¬ ¬ φ → ψ))
Theorem	merlem6 929	Step 12 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
⊢ (χ → (((ψ → χ) → φ) → (θ → φ)))
Theorem	merlem7 930	Between steps 14 and 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
⊢ (φ → (((ψ → χ) → θ) → (((χ → τ) → (¬ θ → ¬ ψ)) → θ)))
Theorem	merlem8 931	Step 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
⊢ (((ψ → χ) → θ) → (((χ → τ) → (¬ θ → ¬ ψ)) → θ))
Theorem	merlem9 932	Step 18 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
⊢ (((φ → ψ) → (χ → (θ → (ψ → τ)))) → (η → (χ → (θ → (ψ → τ)))))
Theorem	merlem10 933	Step 19 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
⊢ ((φ → (φ → ψ)) → (θ → (φ → ψ)))
Theorem	merlem11 934	Step 20 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
⊢ ((φ → (φ → ψ)) → (φ → ψ))
Theorem	merlem12 935	Step 28 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
⊢ (((θ → (¬ ¬ χ → χ)) → φ) → φ)
Theorem	merlem13 936	Step 35 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
⊢ ((φ → ψ) → (((θ → (¬ ¬ χ → χ)) → ¬ ¬ φ) → ψ))
Theorem	luk-1 937	1 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom.
⊢ ((φ → ψ) → ((ψ → χ) → (φ → χ)))
Theorem	luk-2 938	2 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom.
⊢ ((¬ φ → φ) → φ)
Theorem	luk-3 939	3 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom.
⊢ (φ → (¬ φ → ψ))
Theorem	luklem1 940	Used to rederive standard propositional axioms from Lukasiewicz'.
⊢ (φ → ψ)    &   ⊢ (ψ → χ)    ⇒   ⊢ (φ → χ)
Theorem	luklem2 941	Used to rederive standard propositional axioms from Lukasiewicz'.
⊢ ((φ → ¬ ψ) → (((φ → χ) → θ) → (ψ → θ)))
Theorem	luklem3 942	Used to rederive standard propositional axioms from Lukasiewicz'.
⊢ (φ → (((¬ φ → ψ) → χ) → (θ → χ)))
Theorem	luklem4 943	Used to rederive standard propositional axioms from Lukasiewicz'.
⊢ ((((¬ φ → φ) → φ) → ψ) → ψ)
Theorem	luklem5 944	Used to rederive standard propositional axioms from Lukasiewicz'.
⊢ (φ → (ψ → φ))
Theorem	luklem6 945	Used to rederive standard propositional axioms from Lukasiewicz'.
⊢ ((φ → (φ → ψ)) → (φ → ψ))
Theorem	luklem7 946	Used to rederive standard propositional axioms from Lukasiewicz'.
⊢ ((φ → (ψ → χ)) → (ψ → (φ → χ)))
Theorem	luklem8 947	Used to rederive standard propositional axioms from Lukasiewicz'.
⊢ ((φ → ψ) → ((χ → φ) → (χ → ψ)))
Theorem	ax1 948	Standard propositional axiom derived from Lukasiewicz axioms.
⊢ (φ → (ψ → φ))
Theorem	ax2 949	Standard propositional axiom derived from Lukasiewicz axioms.
⊢ ((φ → (ψ → χ)) → ((φ → ψ) → (φ → χ)))
Theorem	ax3 950	Standard propositional axiom derived from Lukasiewicz axioms.
⊢ ((¬ φ → ¬ ψ) → (ψ → φ))
Theorem	nicodraw 951	Axiom of Nicod from Introduction to Mathematical Philosophy B. Russell, p. 152. The axiom is recovered from this raw form by substituting (φ∣ψ) for ¬ (φ ⋀ ψ), where ∣ is the Sheffer stroke (NAND) connective, so that the Sheffer stroke becomes the sole connective. See nicodmpraw 952 for the inference rule. (Based on a proof by Jeff Hoffman, 19-Nov-2007.)
⊢ ¬ (¬ (φ ⋀ ¬ (χ ⋀ ψ)) ⋀ ¬ (¬ (τ ⋀ ¬ (τ ⋀ τ)) ⋀ ¬ (¬ (θ ⋀ χ) ⋀ ¬ (¬ (φ ⋀ θ) ⋀ ¬ (φ ⋀ θ)))))
Theorem	nicodmpraw 952	The inference rule for the axiom of Nicod, in raw form as explained in nicodraw 951.
⊢ φ    &   ⊢ ¬ (φ ⋀ ¬ (χ ⋀ ψ))    ⇒   ⊢ ψ
Predicate calculus axiomatization
The axioms of predicate calculus
Syntax	wal 953	Extend wff definition to include the universal quantifier ('for all'). ∀xφ is read "φ (phi) is true for all x." Typically, in its final application φ would be replaced with a wff containing a (free) occurrence of the variable x, for example x = y. In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of x. When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same.
wff ∀xφ
Syntax	cv 954	This syntax construction states that a variable x, which has been declared to be a set variable by $f statement vx, is also a class expression. This can be justified informally as follows. We know that the class builder {y∣y ∈ x} is a class by cab 1462. Since (when y is distinct from x) we have x = {y∣y ∈ x} by cvjust 1470, we can argue that that the syntax "class x" can be viewed as an abbreviation for "class {y∣y ∈ x}". See the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." While it is tempting and perhaps occasionally useful to view cv 954 as a "type conversion" from a set variable to a class variable, keep in mind that cv 954 is intrinsically no different from any other class-building syntax such as cab 1462, cun 2042, or c0 2277. (The description above applies to set theory, not predicate calculus. The purpose of introducing class x here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 956 of predicate calculus from the wceq 955 of set theory, so that we don't "overload" the = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.)
class x
Syntax	wceq 955	Extend wff definition to include class equality. (The purpose of introducing wff A = B here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 956 of predicate calculus in terms of the wceq 955 of set theory, so that we don't "overload" the = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. For example, some parsers - although not the Metamath program - stumble on the fact that the = in x = y could be the = of either weq 956 or wceq 955, although mathematically it makes no difference. The class variables A and B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 1468 for more information on the set theory usage of wceq 955.)
wff A = B
Theorem	weq 956	Extend wff definition to include atomic formulas using the equality predicate. (Instead of introducing weq 956 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wceq 955. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 956 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 955. Note: To see the proof steps of this syntax proof, type "show proof weq /all" in the Metamath program.)
wff x = y
Syntax	wcel 957	Extend wff definition to include the membership connective between classes. (The purpose of introducing wff A ∈ B here is to allow us to express i.e. "prove" the wel 958 of predicate calculus in terms of the wceq 955 of set theory, so that we don't "overload" the ∈ connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables A and B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-clab 1463 for more information on the set theory usage of wcel 957.)
wff A ∈ B
Theorem	wel 958	Extend wff definition to include atomic formulas with the epsilon (membership) predicate. This is read "x is an element of y," "x is a member of y," "x belongs to y," or "y contains x." Note: The phrase "y includes x" means "x is a subset of y;" to use it also for x ∈ y, as some authors occasionally do, is poor form and causes confusion, according to George Boolos (1992 lecture at MIT). This syntactical construction introduces a binary non-logical predicate symbol ∈ (epsilon) into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for ∈ apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments. (Instead of introducing wel 958 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wcel 957. This lets us avoid overloading the ∈ connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 958 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 957. Note: To see the proof steps of this syntax proof, type "show proof wel /all" in the Metamath program.)
wff x ∈ y
Axiom	ax-5 959	Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105.
⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ))
Axiom	ax-6 960	Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113.
⊢ (¬ ∀xφ → ∀x ¬ ∀xφ)
Axiom	ax-7 961	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 and Axiom C5-3 of [Monk2] p. 113. An alternate axiomatization could use ax467 1022 in place of ax-4 972, ax-6o 977, and ax-7 961.
⊢ (∀x∀yφ → ∀y∀xφ)
Axiom	ax-gen 962	Rule of Generalization. The postulated inference rule of pure predicate calculus. See e.g. Rule 2 of [Hamilton] p. 74. This rule says that if something is unconditionally true, then it is true for all values of a variable. For example, if we have proved x = x, we can conclude ∀xx = x or even ∀yx = x. Theorem a4i 981 shows we can go the other way also: in other words we can add or remove universal quantifiers from the beginning of any theorem as required.
⊢ φ    ⇒   ⊢ ∀xφ
Axiom	ax-8 963	Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. This is similar to, but not quite, a transitive law for equality (proved later as equtr 1130). Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105. Axioms ax-8 963 through ax-16 1209 are the axioms having to do with equality, substitution, and logical properties of our binary predicate ∈ (which later in set theory will mean "is a member of"). Note that all axioms except ax-16 1209 and ax-17 970 are still valid even when x, y, and z are replaced with the same variable because they do not have any distinct variable (Metamath's $d) restrictions. Distinct variable restrictions are required for ax-16 1209 and ax-17 970 only.
⊢ (x = y → (x = z → y = z))
Axiom	ax-9 964	Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. One thing this axiom tells us is that at least one thing exists (although ax-4 972 and possibly others also tell us that, i.e. they are not valid in the empty domain of a "free logic.") In this form (not requiring that x and y be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) It is equivalent to axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint); the equivalence is established by ax9o 1121 and ax9 1123. A more convenient form of this axiom is a9e 1124, which has additional remarks. Raph Levien proved the independence of this axiom from the others on 12-Apr-2005. See item 16 at http://us.metamath.org/award2003.html.
⊢ ¬ ∀x ¬ x = y
Axiom	ax-10 965	Axiom of Quantifier Substitution. One of the equality and substitution axioms of predicate calculus with equality. Appears as Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). This axiom replaces the old axiom ax-10o 1139, which is proved from this one as theorem ax10o 1138. Conversely, this axiom is proved from ax-10o 1139 as theorem ax10 1140.
⊢ (∀x x = y → ∀y y = x)
Axiom	ax-11 966	Axiom of Variable Substitution. One of the 5 equality axioms of predicate calculus. The final consequent ∀x(x = y → φ) is a way of expressing "y substituted for x in wff φ" (cf. sb6 1266). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases. The original version of this axiom, that isn't otherwise used in our development, was ax-11o 1217 ("o" for "old"), which was replaced with this shorter ax-11 966 in Jan. 2007. Juha Arpiainen proved the independence of this axiom (in the form of the older axiom ax-11o 1217) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html. Interestingly, if the wff expression substituted for φ contains no wff variables, the resulting statement can be proved without invoking this axiom. This means that even though this axiom is metalogically independent from the others, it is not logically independent. Specifically, we can prove any wff-variable-free instance of axiom ax-11o 1217 (from which the ax-11 966 instance follows by theorem ax11 1218.) The proof is by induction on formula length, using ax11eq 1362 and ax11el 1363 for the basis steps and ax11indn 1365, ax11indi 1366, and ax11inda 1370 for the induction steps. See also ax11v 1264 and ax11v2 1214 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.
⊢ (x = y → (∀yφ → ∀x(x = y → φ)))
Axiom	ax-12 967	Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever z is distinct from x and y, and x = y is true, then x = y quantified with z is also true. In other words, z is irrelevant to the truth of x = y. Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases. An open problem is whether this axiom is redundant. Note that the analogous axiom for the membership connective, ax-15 1359, has been shown to be redundant. It is also unknown whether this axiom can be replaced by a shorter formula. However, it can be derived from two slightly shorter formulas, as shown by a12study 1377.
⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y)))
Axiom	ax-13 968	Axiom of Equality. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of the ∈ binary predicate. Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be.
⊢ (x = y → (x ∈ z → y ∈ z))
Axiom	ax-14 969	Axiom of Equality. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of the ∈ binary predicate. Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77.
⊢ (x = y → (z ∈ x → z ∈ y))
Axiom	ax-17 970	Axiom to quantify a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113. If we allow the otherwise redundant ax-15 1359 and ax-16 1209, this axiom is logically redundant since it is a metatheorem justified by induction on the number of primitive connectives in wff φ, using ax17eq 1210 and ax17el 1360 together hbn 1003, hbal 1004, and hbim 1006. However, if we omit this axiom our development would be quite inconvenient since we could work only with specific instances of wffs containing no wff variables - this axiom introduces the concept of a set variable not occurring in a wff (as opposed to just two set variables being distinct).
⊢ (φ → ∀xφ)
Derive ax-4, ax-5o, and ax-6o
Theorem	ax4 971	Theorem showing that ax-4 972 can be derived from ax-5 959, ax-gen 962, ax-8 963, ax-9 964, ax-11 966, and ax-17 970 and is therefore redundant. Lemma 21 of [Monk2] p. 114. This theorem should not be referenced in any proof. Instead, we will use ax-4 972 below so that uses of ax-4 972 can be more easily identified. In particular, this will more cleanly separate out the theorems of "pure" predicate calculus that don't involve equality or distinct variables. A beginner can accept ax-4 972 a priori, so that the proof of this theorem (ax4 971), which involves equality as well as the distinct the distinct variable requirements of ax-17 970, can be put off until later. Note: All predicate calculus axioms introduced from this point forward are redundant. Immediately before their introduction, we prove them from earlier axioms to demonstrate their redundancy. Specifically, redundant axioms ax-4 972, ax-5o 974, ax-6o 977, ax-9o 1122, ax-10o 1139, ax-11o 1217, ax-15 1359, and ax-16 1209 are proved by theorems ax4 971, ax5o 973, ax6o 976, ax9o 1121, ax10o 1138, ax11o 1216, ax15 1358, and ax16 1208.
⊢ (∀xφ → φ)
Axiom	ax-4 972	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 φ). (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 ∀xx = y.) This is one of the 4 axioms of what we call "pure" predicate calculus. 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 962. Conditional forms of the converse are given by ax-12 967, ax-15 1359, ax-16 1209, and ax-17 970. Unlike the more general 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 1184. An alternate axiomatization could use ax467 1022 and ax-5o 974 in place of ax-4 972, ax-5o 974, ax-6o 977, and ax-7 961. This axiom is redundant, as shown by theorem ax4 971.
⊢ (∀xφ → φ)
Theorem	ax5o 973	Show that the original axiom ax-5o 974 can be derived from ax-5 959 and others. See ax5 975 for the rederivation of ax-5 959 from ax-5o 974. Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. This theorem should not be referenced in any proof. Instead, use ax-5o 974 below so that uses of ax-5o 974 can be more easily identified.
⊢ (∀x(∀xφ → ψ) → (∀xφ → ∀xψ))
Axiom	ax-5o 974	Axiom of Quantified Implication. This axiom moves a quantifier from outside to inside an implication, quantifying ψ. Notice that x must not be a free variable in the antecedent of the quantified implication, and we express this by binding φ to "protect" the axiom from a φ 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. This axiom is redundant, as shown by theorem ax5o 973.
⊢ (∀x(∀xφ → ψ) → (∀xφ → ∀xψ))
Theorem	ax5 975	Rederivation of axiom ax-5 959 from the orginal version, ax-5o 974. See ax5o 973 for the derivation of ax-5o 974 from ax-5 959. This theorem should not be referenced in any proof. Instead, use ax-5 959 above so that uses of ax-5 959 can be more easily identified.
⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ))
Theorem	ax6o 976	Show that the original axiom ax-6o 977 can be derived from ax-6 960 and others. See ax6 978 for the rederivation of ax-6 960 from ax-6o 977. This theorem should not be referenced in any proof. Instead, use ax-6o 977 below so that uses of ax-6o 977 can be more easily identified.
⊢ (¬ ∀x ¬ ∀xφ → φ)
Axiom	ax-6o 977	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). An alternate axiomatization could use ax467 1022 in place of ax-4 972, ax-6o 977, and ax-7 961. This axiom is redundant, as shown by theorem ax6o 976.
⊢ (¬ ∀x ¬ ∀xφ → φ)
Theorem	ax6 978	Rederivation of axiom ax-6 960 from the orginal version, ax-6o 977. See ax6o 976 for the derivation of ax-6o 977 from ax-6 960. This theorem should not be referenced in any proof. Instead, use ax-6 960 above so that uses of ax-6 960 can be more easily identified.
⊢ (¬ ∀xφ → ∀x ¬ ∀xφ)
Predicate calculus without distinct variables
"Pure" predicate calculus ax-4, ax-5o, ax-6o, ax-gen
Syntax	wex 979	Extend wff definition to include the existential quantifier ("there exists").
wff ∃xφ
Definition	df-ex 980	Define existential quantification. ∃xφ means "there exists at least one set x such that φ is true." Definition of [Margaris] p. 49.
⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
Theorem	a4i 981	Inference rule reversing generalization.
⊢ ∀xφ    ⇒   ⊢ φ
Theorem	gen2 982	Generalization applied twice.
⊢ φ    ⇒   ⊢ ∀x∀yφ
Theorem	a4s 983	Generalization of antecedent.
⊢ (φ → ψ)    ⇒   ⊢ (∀xφ → ψ)
Theorem	a4sd 984	Deduction generalizing antecedent.
⊢ (φ → (ψ → χ))    ⇒   ⊢ (φ → (∀xψ → χ))
Theorem	mpg 985	Modus ponens combined with generalization.
⊢ (∀xφ → ψ)    &   ⊢ φ    ⇒   ⊢ ψ
Theorem	mpgbi 986	Modus ponens on biconditional combined with generalization.
⊢ (∀xφ ↔ ψ)    &   ⊢ φ    ⇒   ⊢ ψ
Theorem	mpgbir 987	Modus ponens on biconditional combined with generalization.
⊢ (φ ↔ ∀xψ)    &   ⊢ ψ    ⇒   ⊢ φ
Theorem	a5i 988	Inference from ax-5o 974.
⊢ (∀xφ → ψ)    ⇒   ⊢ (∀xφ → ∀xψ)
Theorem	a6e 989	Abbreviated version of ax-6o 977.
⊢ (∃x∀xφ → φ)
Theorem	a7s 990	Swap quantifiers in an antecedent.
⊢ (∀x∀yφ → ψ)    ⇒   ⊢ (∀y∀xφ → ψ)
Theorem	19.20i 991	Inference quantifying both antecedent and consequent.
⊢ (φ → ψ)    ⇒   ⊢ (∀xφ → ∀xψ)
Theorem	19.20i2 992	Inference doubly quantifying both antecedent and consequent.
⊢ (φ → ψ)    ⇒   ⊢ (∀x∀yφ → ∀x∀yψ)
Theorem	19.20 993	Theorem 19.20 of [Margaris] p. 90. (The proof was shortened by O'Cat, 30-Mar-2008.)
⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ))
Theorem	19.20ii 994	Inference quantifying antecedent, nested antecedent, and consequent.
⊢ (φ → (ψ → χ))    ⇒   ⊢ (∀xφ → (∀xψ → ∀xχ))
Theorem	19.20d 995	Deduction from Theorem 19.20 of [Margaris] p. 90.
⊢ (φ → ∀xφ)    &   ⊢ (φ → (ψ → χ))    ⇒   ⊢ (φ → (∀xψ → ∀xχ))
Theorem	19.15 996	Theorem 19.15 of [Margaris] p. 90.
⊢ (∀x(φ ↔ ψ) → (∀xφ ↔ ∀xψ))
Theorem	19.21ai 997	Inference from Theorem 19.21 of [Margaris] p. 90.
⊢ (φ → ∀xφ)    &   ⊢ (φ → ψ)    ⇒   ⊢ (φ → ∀xψ)
Theorem	albii 998	Inference adding universal quantifier to both sides of an equivalence.
⊢ (φ ↔ ψ)    ⇒   ⊢ (∀xφ ↔ ∀xψ)
Theorem	2albii 999	Inference adding 2 universal quantifiers to both sides of an equivalence.
⊢ (φ ↔ ψ)    ⇒   ⊢ (∀x∀yφ ↔ ∀x∀yψ)