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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cbvalw 1701* | Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |
⊢ (∀xφ → ∀y∀xφ) & ⊢ (¬ ψ → ∀x ¬ ψ) & ⊢ (∀yψ → ∀x∀yψ) & ⊢ (¬ φ → ∀y ¬ φ) & ⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∀xφ ↔ ∀yψ) | ||
Theorem | cbvalvw 1702* | Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∀xφ ↔ ∀yψ) | ||
Theorem | cbvexvw 1703* | Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∃xφ ↔ ∃yψ) | ||
Theorem | alcomiw 1704* | Weak version of alcom 1737. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) |
⊢ (y = z → (φ ↔ ψ)) ⇒ ⊢ (∀x∀yφ → ∀y∀xφ) | ||
Theorem | hbn1fw 1705* | Weak version of ax-6 1729 from which we can prove any ax-6 1729 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) |
⊢ (∀xφ → ∀y∀xφ) & ⊢ (¬ ψ → ∀x ¬ ψ) & ⊢ (∀yψ → ∀x∀yψ) & ⊢ (¬ φ → ∀y ¬ φ) & ⊢ (¬ ∀yψ → ∀x ¬ ∀yψ) & ⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (¬ ∀xφ → ∀x ¬ ∀xφ) | ||
Theorem | hbn1w 1706* | Weak version of hbn1 1730. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (¬ ∀xφ → ∀x ¬ ∀xφ) | ||
Theorem | hba1w 1707* | Weak version of hba1 1786. See comments for ax6w 1717. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∀xφ → ∀x∀xφ) | ||
Theorem | hbe1w 1708* | Weak version of hbe1 1731. See comments for ax6w 1717. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∃xφ → ∀x∃xφ) | ||
Theorem | hbalw 1709* | Weak version of hbal 1736. Uses only Tarski's FOL axiom schemes. Unlike hbal 1736, this theorem requires that x and y be distinct i.e. are not bundled. (Contributed by NM, 19-Apr-2017.) |
⊢ (x = z → (φ ↔ ψ)) & ⊢ (φ → ∀xφ) ⇒ ⊢ (∀yφ → ∀x∀yφ) | ||
Syntax | wcel 1710 |
Extend wff definition to include the membership connective between
classes.
For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (The purpose of introducing wff A ∈ B here is to allow us to express i.e. "prove" the wel 1711 of predicate calculus in terms of the wceq 1642 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 2340 for more information on the set theory usage of wcel 1710.) |
wff A ∈ B | ||
Theorem | wel 1711 |
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 1711 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 1710. This lets us avoid overloading the ∈ connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1711 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1710. Note: To see the proof steps of this syntax proof, type "show proof wel /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.) |
wff x ∈ y | ||
Axiom | ax-13 1712 | Axiom of Left Equality for Binary Predicate. 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 an arbitrary binary predicate ∈, which we will use for the set membership relation when set theory is introduced. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). "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. (Contributed by NM, 5-Aug-1993.) |
⊢ (x = y → (x ∈ z → y ∈ z)) | ||
Theorem | elequ1 1713 | An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.) |
⊢ (x = y → (x ∈ z ↔ y ∈ z)) | ||
Axiom | ax-14 1714 | Axiom of Right Equality for Binary Predicate. 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 an arbitrary binary predicate ∈, which we will use for the set membership relation when set theory is introduced. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.) |
⊢ (x = y → (z ∈ x → z ∈ y)) | ||
Theorem | elequ2 1715 | An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.) |
⊢ (x = y → (z ∈ x ↔ z ∈ y)) | ||
The orginal axiom schemes of Tarski's predicate calculus are ax-5 1557, ax-17 1616, ax9v 1655, ax-8 1675, ax-13 1712, and ax-14 1714, together with rule ax-gen 1546. See http://us.metamath.org/mpeuni/mmset.html#compare 1546. They are given as axiom schemes B4 through B8 in [KalishMontague] p. 81. These are shown to be logically complete by Theorem 1 of [KalishMontague] p. 85. The axiom system of set.mm includes the auxiliary axiom schemes ax-6 1729, ax-7 1734, ax-12 1925, and ax-11 1746, which are not part of Tarski's axiom schemes. They are used (and we conjecture are required) to make our system "metalogically complete" i.e. able to prove directly all possible schemes with wff and set metavariables, bundled or not, whose object-language instances are valid. (ax-11 1746 has been proved to be required; see http://us.metamath.org/award2003.html#9a. Metalogical independence of the other three are open problems.) (There are additional predicate calculus axiom schemes included in set.mm such as ax-4 2135, but they can all be proved as theorems from the above.) Terminology: Two set (individual) metavariables are "bundled" in an axiom or theorem scheme when there is no distinct variable constraint ($d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the x and y in ax9 1949 are bundled, but they are not in ax9v 1655. We also say that a scheme is bundled when it has at least one pair of bundled set metavariables. If distinct variable conditions are added to all set metavariable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax9v 1655 is the principal instance of ax9 1949. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance ¬ ∀x¬ x = x of ax9 1949 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions ($d) to be concerned with. There is also a small economy in being able to state principal and degenerate instances simultaneously. A disadvantage is that bundling may present difficulties in translations to other proof languages, which typically lack the concept (in part because their variables often represent the variables of the object language rather than metavariables ranging over them). Because Tarski's axiom schemes are logically complete, they can be used to prove any object-language instance of ax-6 1729, ax-7 1734, ax-11 1746, and ax-12 1925 . "Translating" this to Metamath, it means that Tarski's axioms can prove any substitution instance of ax-6 1729, ax-7 1734, ax-11 1746, or ax-12 1925 in which (1) there are no wff metavariables and (2) all set metavariables are mutually distinct i.e. are not bundled. In effect this is mimicking the object language by pretending that each set metavariable is an object-language variable. (There may also be specific instances with wff metavariables and/or bundling that are directly provable from Tarski's axiom schemes, but it isn't guaranteed. Whether all of them are possible is part of the still open metalogical independence problem for our additional axiom schemes.) It can be useful to see how this can be done, both to show that our additional schemes are valid metatheorems of Tarski's system and to be able to translate object language instances of our proofs into proofs that would work with a system using only Tarski's original schemes. In addition, it may (or may not) provide insight into the conjectured metalogical independence of our additional schemes. The new theorem schemes ax6w 1717, ax7w 1718, ax11w 1721, and ax12w 1724 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-6 1729, ax-7 1734, ax-11 1746, and ax-12 1925 meeting conditions (1) and (2). (The "w" suffix stands for "weak version".) Each hypothesis of ax6w 1717, ax7w 1718, and ax11w 1721 is of the form (x = y → (φ ↔ ψ)) where ψ is an auxiliary or "dummy" wff metavariable in which x doesn't occur. We can show by induction on formula length that the hypotheses can be eliminated in all cases meeting conditions (1) and (2). The example ax11wdemo 1723 illustrates the techniques (equality theorems and bound variable renaming) used to achieve this. We also show the degenerate instances for axioms with bundled variables in ax7dgen 1719, ax11dgen 1722, ax12dgen1 1725, ax12dgen2 1726, ax12dgen3 1727, and ax12dgen4 1728. (Their proofs are trivial, but we include them to be thorough.) Combining the principal and degenerate cases outside of Metamath, we show that the bundled schemes ax-6 1729, ax-7 1734, ax-11 1746, and ax-12 1925 are schemes of Tarski's system, meaning that all object language instances they generate are theorems of Tarski's system. It is interesting that Tarski used the bundled scheme ax-9 1654 in an older system, so it seems the main purpose of his later ax9v 1655 was just to show that the weaker unbundled form is sufficient rather than an aesthetic objection to bundled free and bound variables. Since we adopt the bundled ax-9 1654 as our official axiom, we show that the degenerate instance holds in ax9dgen 1716. The case of sp 1747 is curious: originally an axiom of Tarski's system, it was proved redundant by Lemma 9 of [KalishMontague] p. 86. However, the proof is by induction on formula length, and the compact scheme form ∀xφ → φ apparently cannot be proved directly from Tarski's other axioms. The best we can do seems to be spw 1694, again requiring substitution instances of φ that meet conditions (1) and (2) above. Note that our direct proof sp 1747 requires ax-11 1746, which is not part of Tarski's system. | ||
Theorem | ax9dgen 1716 | Tarski's system uses the weaker ax9v 1655 instead of the bundled ax-9 1654, so here we show that the degenerate case of ax-9 1654 can be derived. (Contributed by NM, 23-Apr-2017.) |
⊢ ¬ ∀x ¬ x = x | ||
Theorem | ax6w 1717* | Weak version of ax-6 1729 from which we can prove any ax-6 1729 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (¬ ∀xφ → ∀x ¬ ∀xφ) | ||
Theorem | ax7w 1718* | Weak version of ax-7 1734 from which we can prove any ax-7 1734 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-7 1734, this theorem requires that x and y be distinct i.e. are not bundled. (Contributed by NM, 10-Apr-2017.) |
⊢ (y = z → (φ ↔ ψ)) ⇒ ⊢ (∀x∀yφ → ∀y∀xφ) | ||
Theorem | ax7dgen 1719 | Degenerate instance of ax-7 1734 where bundled variables x and y have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |
⊢ (∀x∀xφ → ∀x∀xφ) | ||
Theorem | ax11wlem 1720* | Lemma for weak version of ax-11 1746. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax11w 1721. (Contributed by NM, 10-Apr-2017.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (x = y → (φ → ∀x(x = y → φ))) | ||
Theorem | ax11w 1721* | Weak version of ax-11 1746 from which we can prove any ax-11 1746 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. An instance of the first hypothesis will normally require that x and y be distinct (unless x does not occur in φ). (Contributed by NM, 10-Apr-2017.) |
⊢ (x = y → (φ ↔ ψ)) & ⊢ (y = z → (φ ↔ χ)) ⇒ ⊢ (x = y → (∀yφ → ∀x(x = y → φ))) | ||
Theorem | ax11dgen 1722 | Degenerate instance of ax-11 1746 where bundled variables x and y have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |
⊢ (x = x → (∀xφ → ∀x(x = x → φ))) | ||
Theorem | ax11wdemo 1723* | Example of an application of ax11w 1721 that results in an instance of ax-11 1746 for a contrived formula with mixed free and bound variables, (x ∈ y ∧ ∀xz ∈ x ∧ ∀y∀zy ∈ x), in place of φ. The proof illustrates bound variable renaming with cbvalvw 1702 to obtain fresh variables to avoid distinct variable clashes. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 14-Apr-2017.) |
⊢ (x = y → (∀y(x ∈ y ∧ ∀x z ∈ x ∧ ∀y∀z y ∈ x) → ∀x(x = y → (x ∈ y ∧ ∀x z ∈ x ∧ ∀y∀z y ∈ x)))) | ||
Theorem | ax12w 1724* | Weak version (principal instance) of ax-12 1925. (Because y and z don't need to be distinct, this actually bundles the principal instance and the degenerate instance (¬ x = y → (y = y → ∀xy = y)).) Uses only Tarski's FOL axiom schemes. The proof is trivial but is included to complete the set ax6w 1717, ax7w 1718, and ax11w 1721. (Contributed by NM, 10-Apr-2017.) |
⊢ (¬ x = y → (y = z → ∀x y = z)) | ||
Theorem | ax12dgen1 1725 | Degenerate instance of ax-12 1925 where bundled variables x and y have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |
⊢ (¬ x = x → (x = z → ∀x x = z)) | ||
Theorem | ax12dgen2 1726 | Degenerate instance of ax-12 1925 where bundled variables x and z have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |
⊢ (¬ x = y → (y = x → ∀x y = x)) | ||
Theorem | ax12dgen3 1727 | Degenerate instance of ax-12 1925 where bundled variables y and z have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) |
⊢ (¬ x = y → (y = y → ∀x y = y)) | ||
Theorem | ax12dgen4 1728 | Degenerate instance of ax-12 1925 where bundled variables x, y, and z have a common substitution. Uses only Tarski's FOL axiom schemes . (Contributed by NM, 13-Apr-2017.) |
⊢ (¬ x = x → (x = x → ∀x x = x)) | ||
In this section we introduce four additional schemes ax-6 1729, ax-7 1734, ax-11 1746, and ax-12 1925 that are not part of Tarski's system but can be proved (outside of Metamath) as theorem schemes of Tarski's system. These are needed to give our system the property of "metalogical completeness," which means that we can prove (with Metamath) all possible schemes expressible in our language of wff metavariables ranging over object-language wffs and set metavariables ranging over object-language individual variables. To show that these schemes are valid metatheorems of Tarski's system S2, above we proved from Tarski's system theorems ax6w 1717, ax7w 1718, ax12w 1724, and ax11w 1721, which show that any object-language instance of these schemes (emulated by having no wff metavariables and requiring all set metavariables to be mutually distinct) can be proved using only the schemes in Tarski's system S2. An open problem is to show that these four additional schemes are metalogically independent from Tarski's. So far, independence of ax-11 1746 from all others has been shown, and independence of Tarski's ax-9 1654 from all others has been shown. | ||
Axiom | ax-6 1729 | Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax6w 1717) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 5-Aug-1993.) |
⊢ (¬ ∀xφ → ∀x ¬ ∀xφ) | ||
Theorem | hbn1 1730 | x is not free in ¬ ∀xφ. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.) |
⊢ (¬ ∀xφ → ∀x ¬ ∀xφ) | ||
Theorem | hbe1 1731 | x is not free in ∃xφ. (Contributed by NM, 5-Aug-1993.) |
⊢ (∃xφ → ∀x∃xφ) | ||
Theorem | nfe1 1732 | x is not free in ∃xφ. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎx∃xφ | ||
Theorem | modal-5 1733 | The analog in our "pure" predicate calculus of axiom 5 of modal logic S5. (Contributed by NM, 5-Oct-2005.) |
⊢ (¬ ∀x ¬ φ → ∀x ¬ ∀x ¬ φ) | ||
Axiom | ax-7 1734 | 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. This axiom scheme is logically redundant (see ax7w 1718) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀x∀yφ → ∀y∀xφ) | ||
Theorem | a7s 1735 | Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀x∀yφ → ψ) ⇒ ⊢ (∀y∀xφ → ψ) | ||
Theorem | hbal 1736 | If x is not free in φ, it is not free in ∀yφ. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (∀yφ → ∀x∀yφ) | ||
Theorem | alcom 1737 | Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀x∀yφ ↔ ∀y∀xφ) | ||
Theorem | alrot3 1738 | Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∀x∀y∀zφ ↔ ∀y∀z∀xφ) | ||
Theorem | alrot4 1739 | Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.) |
⊢ (∀x∀y∀z∀wφ ↔ ∀z∀w∀x∀yφ) | ||
Theorem | hbald 1740 | Deduction form of bound-variable hypothesis builder hbal 1736. (Contributed by NM, 2-Jan-2002.) |
⊢ (φ → ∀yφ) & ⊢ (φ → (ψ → ∀xψ)) ⇒ ⊢ (φ → (∀yψ → ∀x∀yψ)) | ||
Theorem | excom 1741 | Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen to remove dependency on ax-11 1746 ax-6 1729 ax-9 1654 ax-8 1675 and ax-17 1616, 8-Jan-2018.) |
⊢ (∃x∃yφ ↔ ∃y∃xφ) | ||
Theorem | excomim 1742 | One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Revised by Wolf Lammen to remove dependency on ax-11 1746 ax-6 1729 ax-9 1654 ax-8 1675 and ax-17 1616, 8-Jan-2018.) |
⊢ (∃x∃yφ → ∃y∃xφ) | ||
Theorem | excom13 1743 | Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.) |
⊢ (∃x∃y∃zφ ↔ ∃z∃y∃xφ) | ||
Theorem | exrot3 1744 | Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.) |
⊢ (∃x∃y∃zφ ↔ ∃y∃z∃xφ) | ||
Theorem | exrot4 1745 | Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.) |
⊢ (∃x∃y∃z∃wφ ↔ ∃z∃w∃x∃yφ) | ||
Axiom | ax-11 1746 |
Axiom of 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 2099).
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 was ax-11o 2141 ("o" for "old") and was replaced with this shorter ax-11 1746 in Jan. 2007. The old axiom is proved from this one as theorem ax11o 1994. Conversely, this axiom is proved from ax-11o 2141 as theorem ax11 2155. Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-11o 2141) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html. See ax11v 2096 and ax11v2 1992 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions. This axiom scheme is logically redundant (see ax11w 1721) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 22-Jan-2007.) |
⊢ (x = y → (∀yφ → ∀x(x = y → φ))) | ||
Theorem | sp 1747 |
Specialization. A universally quantified wff implies the wff without a
quantifier Axiom scheme B5 of [Tarski] p.
67 (under his system S2,
defined in the last paragraph on p. 77). Also appears as Axiom scheme
C5' in [Megill] p. 448 (p. 16 of the
preprint).
For the axiom of specialization presented in many logic textbooks, see theorem stdpc4 2024. This theorem shows that our obsolete axiom ax-4 2135 can be derived from the others. The proof uses ideas from the proof of Lemma 21 of [Monk2] p. 114. It appears that this scheme cannot be derived directly from Tarski's axioms without auxilliary axiom scheme ax-11 1746. It is thought the best we can do using only Tarski's axioms is spw 1694. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Dec-2017.) |
⊢ (∀xφ → φ) | ||
Theorem | spOLD 1748 | Obsolete proof of sp 1747 as of 23-Dec-2017. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (New usage is discouraged.) |
⊢ (∀xφ → φ) | ||
Theorem | ax5o 1749 |
Show that the original axiom ax-5o 2136 can be derived from ax-5 1557
and
others. See ax5 2146 for the rederivation of ax-5 1557
from ax-5o 2136.
Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.) |
⊢ (∀x(∀xφ → ψ) → (∀xφ → ∀xψ)) | ||
Theorem | ax6o 1750 |
Show that the original axiom ax-6o 2137 can be derived from ax-6 1729
and
others. See ax6 2147 for the rederivation of ax-6 1729
from ax-6o 2137.
Normally, ax6o 1750 should be used rather than ax-6o 2137, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.) |
⊢ (¬ ∀x ¬ ∀xφ → φ) | ||
Theorem | a6e 1751 | Abbreviated version of ax6o 1750. (Contributed by NM, 5-Aug-1993.) |
⊢ (∃x∀xφ → φ) | ||
Theorem | modal-b 1752 | The analog in our "pure" predicate calculus of the Brouwer axiom (B) of modal logic S5. (Contributed by NM, 5-Oct-2005.) |
⊢ (φ → ∀x ¬ ∀x ¬ φ) | ||
Theorem | spi 1753 | Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.) |
⊢ ∀xφ ⇒ ⊢ φ | ||
Theorem | sps 1754 | Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ψ) ⇒ ⊢ (∀xφ → ψ) | ||
Theorem | spsd 1755 | Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.) |
⊢ (φ → (ψ → χ)) ⇒ ⊢ (φ → (∀xψ → χ)) | ||
Theorem | 19.8a 1756 | If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ∃xφ) | ||
Theorem | 19.2g 1757 | Theorem 19.2 of [Margaris] p. 89, generalized to use two set variables. (Contributed by O'Cat, 31-Mar-2008.) |
⊢ (∀xφ → ∃yφ) | ||
Theorem | 19.21bi 1758 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ∀xψ) ⇒ ⊢ (φ → ψ) | ||
Theorem | 19.23bi 1759 | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (∃xφ → ψ) ⇒ ⊢ (φ → ψ) | ||
Theorem | nexr 1760 | Inference from 19.8a 1756. (Contributed by Jeff Hankins, 26-Jul-2009.) |
⊢ ¬ ∃xφ ⇒ ⊢ ¬ φ | ||
Theorem | nfr 1761 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) |
⊢ (Ⅎxφ → (φ → ∀xφ)) | ||
Theorem | nfri 1762 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎxφ ⇒ ⊢ (φ → ∀xφ) | ||
Theorem | nfrd 1763 | Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (φ → Ⅎxψ) ⇒ ⊢ (φ → (ψ → ∀xψ)) | ||
Theorem | alimd 1764 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎxφ & ⊢ (φ → (ψ → χ)) ⇒ ⊢ (φ → (∀xψ → ∀xχ)) | ||
Theorem | alrimi 1765 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎxφ & ⊢ (φ → ψ) ⇒ ⊢ (φ → ∀xψ) | ||
Theorem | nfd 1766 | Deduce that x is not free in ψ in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎxφ & ⊢ (φ → (ψ → ∀xψ)) ⇒ ⊢ (φ → Ⅎxψ) | ||
Theorem | nfdh 1767 | Deduce that x is not free in ψ in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ (φ → ∀xφ) & ⊢ (φ → (ψ → ∀xψ)) ⇒ ⊢ (φ → Ⅎxψ) | ||
Theorem | alrimdd 1768 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎxφ & ⊢ (φ → Ⅎxψ) & ⊢ (φ → (ψ → χ)) ⇒ ⊢ (φ → (ψ → ∀xχ)) | ||
Theorem | alrimd 1769 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎxφ & ⊢ Ⅎxψ & ⊢ (φ → (ψ → χ)) ⇒ ⊢ (φ → (ψ → ∀xχ)) | ||
Theorem | eximd 1770 | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎxφ & ⊢ (φ → (ψ → χ)) ⇒ ⊢ (φ → (∃xψ → ∃xχ)) | ||
Theorem | nexd 1771 | Deduction for generalization rule for negated wff. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎxφ & ⊢ (φ → ¬ ψ) ⇒ ⊢ (φ → ¬ ∃xψ) | ||
Theorem | albid 1772 | Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎxφ & ⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∀xψ ↔ ∀xχ)) | ||
Theorem | exbid 1773 | Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎxφ & ⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∃xψ ↔ ∃xχ)) | ||
Theorem | nfbidf 1774 | An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) |
⊢ Ⅎxφ & ⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (Ⅎxψ ↔ Ⅎxχ)) | ||
Theorem | hbnt 1775 | Closed theorem version of bound-variable hypothesis builder hbn 1776. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀x(φ → ∀xφ) → (¬ φ → ∀x ¬ φ)) | ||
Theorem | hbn 1776 | If x is not free in φ, it is not free in ¬ φ. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 17-Dec-2017.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (¬ φ → ∀x ¬ φ) | ||
Theorem | hbnOLD 1777 | Obsolete proof of hbn 1776 as of 16-Dec-2017. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (¬ φ → ∀x ¬ φ) | ||
Theorem | 19.9ht 1778 | A closed version of 19.9 1783. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀x(φ → ∀xφ) → (∃xφ → φ)) | ||
Theorem | 19.9t 1779 | A closed version of 19.9 1783. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen, 30-Dec-2017.) |
⊢ (Ⅎxφ → (∃xφ ↔ φ)) | ||
Theorem | 19.9h 1780 | A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Proof shortened by Wolf Lammen, 5-Jan-2018.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (∃xφ ↔ φ) | ||
Theorem | 19.9hOLD 1781 | Obsolete proof of 19.9h 1780 as of 5-Jan-2018. (Contributed by FL, 24-Mar-2007.) (New usage is discouraged.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (∃xφ ↔ φ) | ||
Theorem | 19.9d 1782 | A deduction version of one direction of 19.9 1783. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
⊢ (ψ → Ⅎxφ) ⇒ ⊢ (ψ → (∃xφ → φ)) | ||
Theorem | 19.9 1783 | A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) |
⊢ Ⅎxφ ⇒ ⊢ (∃xφ ↔ φ) | ||
Theorem | 19.9OLD 1784 | Obsolete proof of 19.9 1783 as of 30-Dec-2017. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.) |
⊢ Ⅎxφ ⇒ ⊢ (∃xφ ↔ φ) | ||
Theorem | 19.3 1785 | A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎxφ ⇒ ⊢ (∀xφ ↔ φ) | ||
Theorem | hba1 1786 | x is not free in ∀xφ. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 15-Dec-2017.) |
⊢ (∀xφ → ∀x∀xφ) | ||
Theorem | hba1OLD 1787 | Obsolete proof of hba1 1786 as of 15-Dec-2017 (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (∀xφ → ∀x∀xφ) | ||
Theorem | nfa1 1788 | x is not free in ∀xφ. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎx∀xφ | ||
Theorem | a5i 1789 | Inference version of ax5o 1749. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀xφ → ψ) ⇒ ⊢ (∀xφ → ∀xψ) | ||
Theorem | nfnf1 1790 | x is not free in Ⅎxφ. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ ℲxℲxφ | ||
Theorem | nfnd 1791 | If in a context x is not free in ψ, it is not free in ¬ ψ. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) |
⊢ (φ → Ⅎxψ) ⇒ ⊢ (φ → Ⅎx ¬ ψ) | ||
Theorem | nfndOLD 1792 | Obsolete proof of nfnd 1791 as of 28-Dec-2017. (Contributed by Mario Carneiro, 24-Sep-2016.) (New usage is discouraged.) |
⊢ (φ → Ⅎxψ) ⇒ ⊢ (φ → Ⅎx ¬ ψ) | ||
Theorem | nfn 1793 | If x is not free in φ, it is not free in ¬ φ. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎxφ ⇒ ⊢ Ⅎx ¬ φ | ||
Theorem | 19.38 1794 | Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen, 2-Jan-2018.) |
⊢ ((∃xφ → ∀xψ) → ∀x(φ → ψ)) | ||
Theorem | 19.21t 1795 | Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) |
⊢ (Ⅎxφ → (∀x(φ → ψ) ↔ (φ → ∀xψ))) | ||
Theorem | 19.21 1796 | Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "x is not free in φ." (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎxφ ⇒ ⊢ (∀x(φ → ψ) ↔ (φ → ∀xψ)) | ||
Theorem | 19.21h 1797 | Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "x is not free in φ." (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (∀x(φ → ψ) ↔ (φ → ∀xψ)) | ||
Theorem | stdpc5 1798 | An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis Ⅎxφ can be thought of as emulating "x is not free in φ." With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example x would not (for us) be free in x = x by nfequid 1678. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) |
⊢ Ⅎxφ ⇒ ⊢ (∀x(φ → ψ) → (φ → ∀xψ)) | ||
Theorem | stdpc5OLD 1799 | Obsolete proof of stdpc5 1798 as of 1-Jan-2018. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎxφ ⇒ ⊢ (∀x(φ → ψ) → (φ → ∀xψ)) | ||
Theorem | 19.23t 1800 | Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
⊢ (Ⅎxψ → (∀x(φ → ψ) ↔ (∃xφ → ψ))) |
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