P. 18 of Theorem List - New Foundations Explorer

Theorem List for New Foundations Explorer - 1701-1800   *Has distinct variable group(s)

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)
1.4.8  Membership predicate
Syntax	wcel 1710	Extend wff definition to include the membership connective between classes. For a general discussion of the theory of classes, see https://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
1.4.9  Axiom schemes ax-13 (Left Equality for Binary Predicate)
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))
1.4.10  Axiom schemes ax-14 (Right Equality for Binary Predicate)
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))
1.4.11  Logical redundancy of ax-6 , ax-7 , ax-11 , ax-12 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 https://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 setvar metavariables, bundled or not, whose object-language instances are valid. (ax-11 1746 has been proved to be required; see https://us.metamath.org/award2003.html#9a 1746. 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 setvar (individual) metavariables are "bundled" in an axiom or theorem scheme when there is no distinct variable constraint ($d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the x and y in ax9 1949 are bundled, but they are not in ax9v 1655. We also say that a scheme is bundled when it has at least one pair of bundled setvar metavariables. If distinct variable conditions are added to all setvar metavariable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax9v 1655 is the principal instance of ax9 1949. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance ¬ ∀x¬ x = x of ax9 1949 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions ($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 setvr metavariables are mutually distinct i.e. are not bundled. In effect this is mimicking the object language by pretending that each setvar 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))
1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes) 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 setvar 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 setvar 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.
1.5.1  Axiom scheme ax-6 (Quantified Negation)
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)
1.5.2  Axiom scheme ax-7 (Quantifier Commutation)
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)
1.5.3  Axiom scheme ax-11 (Substitution)
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 https://us.metamath.org/award2003.html 2141. 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 auxiliary 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.) (Proof modification is discouraged.) (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 setvar 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.) (Proof modification is discouraged.) (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.) (Proof modification is discouraged.) (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.) (Proof modification is discouraged.) (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.) (Proof modification is discouraged.) (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.) (Proof modification is discouraged.) (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.) (Proof modification is discouraged.) (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)))