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Theorem List for Metamath Proof Explorer - 2101-2200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremequsb3r 2101* Substitution applied to the atomic wff with equality. Variant of equsb3 2100. (Contributed by AV, 29-Jul-2023.) (Proof shortened by Wolf Lammen, 2-Sep-2023.)
([𝑦 / 𝑥]𝑧 = 𝑥𝑧 = 𝑦)
 
Theoremequsb3rOLD 2102* Obsolete version of equsb3r 2101 as of 2-Sep-2023. (Contributed by AV, 29-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥]𝑧 = 𝑥𝑧 = 𝑦)
 
Theoremequsb1v 2103* Version of equsb1 2526 with a disjoint variable condition, which neither requires ax-12 2167 nor ax-13 2383. (Contributed by BJ, 11-Sep-2019.) Remove dependencies on axioms. (Revised by Wolf Lammen, 30-May-2023.) (Proof shortened by Steven Nguyen, 19-Jun-2023.) Revise df-sb 2061. (Revised by Steven Nguyen, 11-Jul-2023.) (Proof shortened by Steven Nguyen, 22-Jul-2023.)
[𝑦 / 𝑥]𝑥 = 𝑦
 
Theoremequsb1vOLD 2104* Obsolete version of equsb1v 2103 as of 22-Jul-2023. Version of equsb1 2526 with a disjoint variable condition, which neither requires ax-12 2167 nor ax-13 2383. (Contributed by BJ, 11-Sep-2019.) Remove dependencies on axioms. (Revised by Wolf Lammen, 30-May-2023.) (Proof shortened by Steven Nguyen, 19-Jun-2023.) Revise df-sb 2061. (Revised by Steven Nguyen, 11-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
[𝑦 / 𝑥]𝑥 = 𝑦
 
1.4.9  Membership predicate
 
Syntaxwcel 2105 Extend wff definition to include the membership connective between classes.

For a general discussion of the theory of classes, see mmset.html#class.

(The purpose of introducing wff 𝐴𝐵 here is to allow us to express, i.e., "prove", the wel 2106 of predicate calculus in terms of the wcel 2105 of set theory, so that we do not "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 2800 for more information on the set theory usage of wcel 2105.)

wff 𝐴𝐵
 
Theoremwel 2106 Extend wff definition to include atomic formulas with the 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 𝑥𝑦, as some authors occasionally do, is poor form and causes confusion, according to George Boolos (1992 lecture at MIT).

This syntactic construction introduces a binary non-logical predicate symbol (stylized lowercase 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 2106 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 2105. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 2106 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 2105. Note: To see the proof steps of this syntax proof, type "MM> SHOW PROOF wel / ALL" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

wff 𝑥𝑦
 
1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)
 
Axiomax-8 2107 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.

We prove in ax8 2111 that this axiom can be recovered from its weakened version ax8v 2108 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-8 2107 should be ax8v 2108. See the comment of ax8v 2108 for more details on these matters. (Contributed by NM, 30-Jun-1993.) (Revised by BJ, 7-Dec-2020.) Use ax8 2111 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 
Theoremax8v 2108* Weakened version of ax-8 2107, with a disjoint variable condition on 𝑥, 𝑦. This should be the only proof referencing ax-8 2107, and it should be referenced only by its two weakened versions ax8v1 2109 and ax8v2 2110, from which ax-8 2107 is then rederived as ax8 2111, which shows that either ax8v 2108 or the conjunction of ax8v1 2109 and ax8v2 2110 is sufficient. (Contributed by BJ, 7-Dec-2020.) Use ax8 2111 instead. (New usage is discouraged.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 
Theoremax8v1 2109* First of two weakened versions of ax8v 2108, with an extra disjoint variable condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 
Theoremax8v2 2110* Second of two weakened versions of ax8v 2108, with an extra disjoint variable condition on 𝑦, 𝑧 see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 
Theoremax8 2111 Proof of ax-8 2107 from ax8v1 2109 and ax8v2 2110, proving sufficiency of the conjunction of the latter two weakened versions of ax8v 2108, which is itself a weakened version of ax-8 2107. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 
Theoremelequ1 2112 An identity law for the non-logical predicate. (Contributed by NM, 30-Jun-1993.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 
Theoremelsb3 2113* Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 24-Jul-2023.)
([𝑦 / 𝑥]𝑥𝑧𝑦𝑧)
 
Theoremcleljust 2114* When the class variables in definition df-clel 2893 are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 2106 with the class variables in wcel 2105. (Contributed by NM, 28-Jan-2004.) Revised to use equsexvw 2002 in order to remove dependencies on ax-10 2136, ax-12 2167, ax-13 2383. Note that there is no disjoint variable condition on 𝑥, 𝑦, that is, on the variables of the left-hand side, as should be the case for definitions. (Revised by BJ, 29-Dec-2020.)
(𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
 
1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)
 
Axiomax-9 2115 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).

We prove in ax9 2119 that this axiom can be recovered from its weakened version ax9v 2116 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-9 2115 should be ax9v 2116. See the comment of ax9v 2116 for more details on these matters. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 7-Dec-2020.) Use ax9 2119 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 
Theoremax9v 2116* Weakened version of ax-9 2115, with a disjoint variable condition on 𝑥, 𝑦. This should be the only proof referencing ax-9 2115, and it should be referenced only by its two weakened versions ax9v1 2117 and ax9v2 2118, from which ax-9 2115 is then rederived as ax9 2119, which shows that either ax9v 2116 or the conjunction of ax9v1 2117 and ax9v2 2118 is sufficient. (Contributed by BJ, 7-Dec-2020.) Use ax9 2119 instead. (New usage is discouraged.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 
Theoremax9v1 2117* First of two weakened versions of ax9v 2116, with an extra disjoint variable condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 
Theoremax9v2 2118* Second of two weakened versions of ax9v 2116, with an extra disjoint variable condition on 𝑦, 𝑧 see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 
Theoremax9 2119 Proof of ax-9 2115 from ax9v1 2117 and ax9v2 2118, proving sufficiency of the conjunction of the latter two weakened versions of ax9v 2116, which is itself a weakened version of ax-9 2115. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 
Theoremelequ2 2120 An identity law for the non-logical predicate. (Contributed by NM, 21-Jun-1993.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 
Theoremelsb4 2121* Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 24-Jul-2023.)
([𝑦 / 𝑥]𝑧𝑥𝑧𝑦)
 
Theoremelequ2g 2122* A form of elequ2 2120 with a universal quantifier. Its converse is the axiom of extensionality ax-ext 2793. (Contributed by BJ, 3-Oct-2019.)
(𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))
 
1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13

The original axiom schemes of Tarski's predicate calculus are ax-4 1801, ax-5 1902, ax6v 1962, ax-7 2006, ax-8 2107, and ax-9 2115, together with rule ax-gen 1787. See mmset.html#compare 1787. 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-10 2136, ax-11 2151, ax-12 2167, and ax-13 2383, which are not part of Tarski's axiom schemes. Each object-language instance of them is provable from Tarski's axioms, so they are logically redundant. However, they are conjectured not to be provable directly as schemes from Tarski's axiom schemes using only Metamath's direct substitution rule. They are used to make our system "metalogically complete", i.e., able to prove directly all possible schemes with wff and setvar variables, bundled or not, whose object-language instances are valid. (ax-12 2167 has been proved to be required; see https://us.metamath.org/award2003.html#9a 2167. Metalogical independence of the other three are open problems.)

(There are additional predicate calculus axiom schemes included in set.mm such as ax-c5 35901, 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 ($d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the 𝑥 and 𝑦 in ax-6 1961 are bundled, but they are not in ax6v 1962. We also say that a scheme is bundled when it has at least one pair of bundled setvar variables. If distinct variable conditions are added to all setvar variable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax6v 1962 is the principal instance of ax-6 1961. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance ¬ ∀𝑥¬ 𝑥 = 𝑥 of ax-6 1961 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions ($d) to be concerned with, and theorems are more general. There may be some economy in being able to prove facts about 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-10 2136, ax-11 2151, ax-12 2167, and ax-13 2383. "Translating" this to Metamath, it means that Tarski's axioms can prove any substitution instance of ax-10 2136, ax-11 2151, ax-12 2167, or ax-13 2383 in which (1) there are no wff metavariables and (2) all setvar variables are mutually distinct i.e. are not bundled. In effect this is mimicking the object language by pretending that each setvar variable 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 theorem schemes ax10w 2124, ax11w 2125, ax12w 2128, and ax13w 2131 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-10 2136, ax-11 2151, ax-12 2167, and ax-13 2383 meeting Conditions (1) and (2). (The "w" suffix stands for "weak version".) Each hypothesis of ax10w 2124, ax11w 2125, and ax12w 2128 is of the form (𝑥 = 𝑦 → (𝜑𝜓)) 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 ax12wdemo 2130 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 ax11dgen 2126, ax12dgen 2129, ax13dgen1 2132, ax13dgen2 2133, ax13dgen3 2134, and ax13dgen4 2135. (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-10 2136, ax-11 2151, ax-12 2167, and ax-13 2383 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-6 1961 in an older system, so it seems the main purpose of his later ax6v 1962 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-6 1961 as our official axiom, we show that the degenerate instance holds in ax6dgen 2123. (Recall that in set.mm, the only statement referencing ax-6 1961 is ax6v 1962.)

The case of sp 2172 is curious: originally an axiom scheme of Tarski's system, it was proved logically redundant by Lemma 9 of [KalishMontague] p. 86. However, the proof is by induction on formula length, and the scheme form 𝑥𝜑𝜑 apparently cannot be proved directly from Tarski's other axiom schemes. The best we can do seems to be spw 2032, again requiring substitution instances of 𝜑 that meet Conditions (1) and (2) above. Note that our direct proof sp 2172 requires ax-12 2167, which is not part of Tarski's system.

 
Theoremax6dgen 2123 Tarski's system uses the weaker ax6v 1962 instead of the bundled ax-6 1961, so here we show that the degenerate case of ax-6 1961 can be derived. Even though ax-6 1961 is in the list of axioms used, recall that in set.mm, the only statement referencing ax-6 1961 is ax6v 1962. We later rederive from ax6v 1962 the bundled form as ax6 2395 with the help of the auxiliary axiom schemes. (Contributed by NM, 23-Apr-2017.)
¬ ∀𝑥 ¬ 𝑥 = 𝑥
 
Theoremax10w 2124* Weak version of ax-10 2136 from which we can prove any ax-10 2136 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. It is an alias of hbn1w 2044 introduced for labeling consistency. (Contributed by NM, 9-Apr-2017.) Use hbn1w 2044 instead. (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
 
Theoremax11w 2125* Weak version of ax-11 2151 from which we can prove any ax-11 2151 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 2151, this theorem requires that 𝑥 and 𝑦 be distinct i.e. are not bundled. It is an alias of alcomiw 2041 introduced for labeling consistency. (Contributed by NM, 10-Apr-2017.) Use alcomiw 2041 instead. (New usage is discouraged.)
(𝑦 = 𝑧 → (𝜑𝜓))       (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
Theoremax11dgen 2126 Degenerate instance of ax-11 2151 where bundled variables 𝑥 and 𝑦 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
(∀𝑥𝑥𝜑 → ∀𝑥𝑥𝜑)
 
Theoremax12wlem 2127* Lemma for weak version of ax-12 2167. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax12w 2128. (Contributed by NM, 10-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theoremax12w 2128* Weak version of ax-12 2167 from which we can prove any ax-12 2167 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). For an example of how the hypotheses can be eliminated when we substitute an expression without wff variables for 𝜑, see ax12wdemo 2130. (Contributed by NM, 10-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑦 = 𝑧 → (𝜑𝜒))       (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theoremax12dgen 2129 Degenerate instance of ax-12 2167 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 = 𝑥𝜑)))
 
Theoremax12wdemo 2130* Example of an application of ax12w 2128 that results in an instance of ax-12 2167 for a contrived formula with mixed free and bound variables, (𝑥𝑦 ∧ ∀𝑥𝑧𝑥 ∧ ∀𝑦𝑧𝑦𝑥), in place of 𝜑. The proof illustrates bound variable renaming with cbvalvw 2034 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(𝑥𝑦 ∧ ∀𝑥 𝑧𝑥 ∧ ∀𝑦𝑧 𝑦𝑥) → ∀𝑥(𝑥 = 𝑦 → (𝑥𝑦 ∧ ∀𝑥 𝑧𝑥 ∧ ∀𝑦𝑧 𝑦𝑥))))
 
Theoremax13w 2131* Weak version (principal instance) of ax-13 2383. (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 ax10w 2124, ax11w 2125, and ax12w 2128. (Contributed by NM, 10-Apr-2017.)
𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
 
Theoremax13dgen1 2132 Degenerate instance of ax-13 2383 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))
 
Theoremax13dgen2 2133 Degenerate instance of ax-13 2383 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))
 
Theoremax13dgen3 2134 Degenerate instance of ax-13 2383 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))
 
Theoremax13dgen4 2135 Degenerate instance of ax-13 2383 where bundled variables 𝑥, 𝑦, and 𝑧 have a common substitution. Therefore, also a degenerate instance of ax13dgen1 2132, ax13dgen2 2133, and ax13dgen3 2134. Also an instance of the intuitionistic tautology pm2.21 123. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Oct-2021.)
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-10 2136, ax-11 2151, ax-12 2167, and ax-13 2383 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 "scheme completeness", which means that we can prove (with Metamath) all possible theorem schemes expressible in our language of wff metavariables ranging over object-language wffs, and setvar variables 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 ax10w 2124, ax11w 2125, ax12w 2128, and ax13w 2131, which show that any object-language instance of these schemes (emulated by having no wff metavariables and requiring all setvar variables 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 mutually metalogically independent and metalogically independent from Tarski's. So far, independence of ax-12 2167 from all others has been shown, and independence of Tarski's ax-6 1961 from all others has been shown; see items 9a and 11 on https://us.metamath.org/award2003.html 1961.

 
1.5.1  Axiom scheme ax-10 (Quantified Negation)
 
Axiomax-10 2136 Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax10w 2124) but is used as an auxiliary axiom scheme to achieve scheme completeness. It means that 𝑥 is not free in ¬ ∀𝑥𝜑. (Contributed by NM, 21-May-2008.) Use its alias hbn1 2137 instead if you must use it. Any theorem in first-order logic (FOL) that contains only set variables that are all mutually distinct, and has no wff variables, can be proved *without* using ax-10 2136 through ax-13 2383, by invoking ax10w 2124 through ax13w 2131. We encourage proving theorems *without* ax-10 2136 through ax-13 2383 and moving them up to the ax-4 1801 through ax-9 2115 section. (New usage is discouraged.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
 
Theoremhbn1 2137 Alias for ax-10 2136 to be used instead of it. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
 
Theoremhbe1 2138 The setvar 𝑥 is not free in 𝑥𝜑. Corresponds to the axiom (5) of modal logic (see also modal5 2150). (Contributed by NM, 24-Jan-1993.)
(∃𝑥𝜑 → ∀𝑥𝑥𝜑)
 
Theoremhbe1a 2139 Dual statement of hbe1 2138. Modified version of axc7e 2329 with a universally quantified consequent. (Contributed by Wolf Lammen, 15-Sep-2021.)
(∃𝑥𝑥𝜑 → ∀𝑥𝜑)
 
Theoremnf5-1 2140 One direction of nf5 2282 can be proved with a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 16-Sep-2021.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
 
Theoremnf5i 2141 Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑 → ∀𝑥𝜑)       𝑥𝜑
 
Theoremnf5dh 2142 Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) df-nf 1776 changed. (Revised by Wolf Lammen, 11-Oct-2021.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)
 
Theoremnf5dv 2143* Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1776 changed. (Revised by Wolf Lammen, 18-Sep-2021.) (Proof shortened by Wolf Lammen, 13-Jul-2022.)
(𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)
 
Theoremnfnaew 2144* Version of nfnae 2451 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑧 ¬ ∀𝑥 𝑥 = 𝑦
 
Theoremnfe1 2145 The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝑥𝜑
 
Theoremnfa1 2146 The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1776 changed. (Revised by Wolf Lammen, 11-Sep-2021.) Remove dependency on ax-12 2167. (Revised by Wolf Lammen, 12-Oct-2021.)
𝑥𝑥𝜑
 
Theoremnfna1 2147 A convenience theorem particularly designed to remove dependencies on ax-11 2151 in conjunction with distinctors. (Contributed by Wolf Lammen, 2-Sep-2018.)
𝑥 ¬ ∀𝑥𝜑
 
Theoremnfia1 2148 Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥(∀𝑥𝜑 → ∀𝑥𝜓)
 
Theoremnfnf1 2149 The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2167. (Revised by Wolf Lammen, 12-Oct-2021.)
𝑥𝑥𝜑
 
Theoremmodal5 2150 The analogue in our predicate calculus of axiom (5) of modal logic S5. See also hbe1 2138. (Contributed by NM, 5-Oct-2005.)
(¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
 
1.5.2  Axiom scheme ax-11 (Quantifier Commutation)
 
Axiomax-11 2151 Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. 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 ax11w 2125) but is used as an auxiliary axiom scheme to achieve metalogical completeness. (Contributed by NM, 12-Mar-1993.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
Theoremalcoms 2152 Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.)
(∀𝑥𝑦𝜑𝜓)       (∀𝑦𝑥𝜑𝜓)
 
Theoremalcom 2153 Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 30-Jun-1993.)
(∀𝑥𝑦𝜑 ↔ ∀𝑦𝑥𝜑)
 
Theoremalrot3 2154 Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦𝑧𝜑 ↔ ∀𝑦𝑧𝑥𝜑)
 
Theoremalrot4 2155 Rotate four universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
(∀𝑥𝑦𝑧𝑤𝜑 ↔ ∀𝑧𝑤𝑥𝑦𝜑)
 
Theoremsbal 2156* Move universal quantifier in and out of substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 13-Aug-2023.)
([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
 
Theoremsbalv 2157* Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)
 
Theoremsbcom2 2158* Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 23-Dec-2022.)
([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
 
Theoremexcom 2159 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-5 1902, ax-6 1961, ax-7 2006, ax-10 2136, ax-12 2167. (Revised by Wolf Lammen, 8-Jan-2018.) (Proof shortened by Wolf Lammen, 22-Aug-2020.)
(∃𝑥𝑦𝜑 ↔ ∃𝑦𝑥𝜑)
 
Theoremexcomim 2160 One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Remove dependencies on ax-5 1902, ax-6 1961, ax-7 2006, ax-10 2136, ax-12 2167. (Revised by Wolf Lammen, 8-Jan-2018.)
(∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑)
 
Theoremexcom13 2161 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝜑 ↔ ∃𝑧𝑦𝑥𝜑)
 
Theoremexrot3 2162 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(∃𝑥𝑦𝑧𝜑 ↔ ∃𝑦𝑧𝑥𝜑)
 
Theoremexrot4 2163 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑧𝑤𝑥𝑦𝜑)
 
Theoremhbal 2164 If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by NM, 12-Mar-1993.)
(𝜑 → ∀𝑥𝜑)       (∀𝑦𝜑 → ∀𝑥𝑦𝜑)
 
Theoremhbald 2165 Deduction form of bound-variable hypothesis builder hbal 2164. (Contributed by NM, 2-Jan-2002.)
(𝜑 → ∀𝑦𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → (∀𝑦𝜓 → ∀𝑥𝑦𝜓))
 
Theoremnfa2 2166 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2167. (Revised by Wolf Lammen, 18-Oct-2021.)
𝑥𝑦𝑥𝜑
 
1.5.3  Axiom scheme ax-12 (Substitution)
 
Axiomax-12 2167 Axiom of Substitution. One of the 5 equality axioms of predicate calculus. The final consequent 𝑥(𝑥 = 𝑦𝜑) is a way of expressing "𝑦 substituted for 𝑥 in wff 𝜑 " (cf. sb6 2084). 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-c15 35907 and was replaced with this shorter ax-12 2167 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2438. Conversely, this axiom is proved from ax-c15 35907 as theorem ax12 2440.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 35907) from the others on 19-Jan-2006. See item 9a at https://us.metamath.org/award2003.html 35907.

See ax12v 2168 and ax12v2 2169 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax12w 2128) but is used as an auxiliary axiom scheme to achieve scheme completeness. (Contributed by NM, 22-Jan-2007.) (New usage is discouraged.)

(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theoremax12v 2168* This is essentially axiom ax-12 2167 weakened by additional restrictions on variables. Besides axc11r 2379, this theorem should be the only one referencing ax-12 2167 directly.

Both restrictions on variables have their own value. If for a moment we assume 𝑥 could be set to 𝑦, then, after elimination of the tautology 𝑦 = 𝑦, immediately we have 𝜑 → ∀𝑦𝜑 for all 𝜑 and 𝑦, that is ax-5 1902, a degenerate result.

The second restriction is not necessary, but a simplification that makes the following interpretation easier to see. Since 𝜑 textually at most depends on 𝑥, we can look at it at some given 'fixed' 𝑦. This theorem now states that the truth value of 𝜑 will stay constant, as long as we 'vary 𝑥 around 𝑦' only such that 𝑥 = 𝑦 still holds. Or in other words, equality is the finest grained logical expression. If you cannot differ two sets by =, you won't find a whatever sophisticated expression that does. One might wonder how the described variation of 𝑥 is possible at all. Note that Metamath is a text processor that easily sees a difference between text chunks {𝑥 ∣ ¬ 𝑥 = 𝑥} and {𝑦 ∣ ¬ 𝑦 = 𝑦}. Our usual interpretation is to abstract from textual variations of the same set, but we are free to interpret Metamath's formalism differently, and in fact let 𝑥 run through all textual representations of sets.

Had we allowed 𝜑 to depend also on 𝑦, this idea is both harder to see, and it is less clear that this extra freedom introduces effects not covered by other axioms. (Contributed by Wolf Lammen, 8-Aug-2020.)

(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theoremax12v2 2169* It is possible to remove any restriction on 𝜑 in ax12v 2168. Same as Axiom C8 of [Monk2] p. 105. Use ax12v 2168 instead when sufficient. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-10 2136 and ax-13 2383. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theorem19.8a 2170 If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. See 19.8v 1978 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 9-Jan-1993.) Allow a shortening of sp 2172. (Revised by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
(𝜑 → ∃𝑥𝜑)
 
Theorem19.8ad 2171 If a wff is true, it is true for at least one instance. Deduction form of 19.8a 2170. (Contributed by DAW, 13-Feb-2017.)
(𝜑𝜓)       (𝜑 → ∃𝑥𝜓)
 
Theoremsp 2172 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). This corresponds to the axiom (T) of modal logic.

For the axiom of specialization presented in many logic textbooks, see theorem stdpc4 2064.

This theorem shows that our obsolete axiom ax-c5 35901 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-12 2167. It is thought the best we can do using only Tarski's axioms is spw 2032. Also see spvw 1976 where 𝑥 and 𝜑 are disjoint, using fewer axioms. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)

(∀𝑥𝜑𝜑)
 
Theoremspi 2173 Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1787. Contrary to the rule of generalization, its closed form is valid, see sp 2172. (Contributed by NM, 5-Aug-1993.)
𝑥𝜑       𝜑
 
Theoremsps 2174 Generalization of antecedent. (Contributed by NM, 5-Jan-1993.)
(𝜑𝜓)       (∀𝑥𝜑𝜓)
 
Theorem2sp 2175 A double specialization (see sp 2172). Another double specialization, closer to PM*11.1, is 2stdpc4 2066. (Contributed by BJ, 15-Sep-2018.)
(∀𝑥𝑦𝜑𝜑)
 
Theoremspsd 2176 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))
 
Theorem19.2g 2177 Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. Use 19.2 1972 when sufficient. (Contributed by Mel L. O'Cat, 31-Mar-2008.)
(∀𝑥𝜑 → ∃𝑦𝜑)
 
Theorem19.21bi 2178 Inference form of 19.21 2198 and also deduction form of sp 2172. (Contributed by NM, 26-May-1993.)
(𝜑 → ∀𝑥𝜓)       (𝜑𝜓)
 
Theorem19.21bbi 2179 Inference removing two universal quantifiers. Version of 19.21bi 2178 with two quantifiers. (Contributed by NM, 20-Apr-1994.)
(𝜑 → ∀𝑥𝑦𝜓)       (𝜑𝜓)
 
Theorem19.23bi 2180 Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2202. (Contributed by NM, 12-Mar-1993.)
(∃𝑥𝜑𝜓)       (𝜑𝜓)
 
Theoremnexr 2181 Inference associated with the contrapositive of 19.8a 2170. (Contributed by Jeff Hankins, 26-Jul-2009.)
¬ ∃𝑥𝜑        ¬ 𝜑
 
Theoremqexmid 2182 Quantified excluded middle (see exmid 888). Also known as the drinker paradox (if 𝜑(𝑥) is interpreted as "𝑥 drinks", then this theorem tells that there exists a person such that, if this person drinks, then everyone drinks). Exercise 9.2a of Boolos, p. 111, Computability and Logic. (Contributed by NM, 10-Dec-2000.)
𝑥(𝜑 → ∀𝑥𝜑)
 
Theoremnf5r 2183 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1776 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by Wolf Lammen, 23-Nov-2023.)
(Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
 
Theoremnf5rOLD 2184 Obsolete version of nfrd 1783 as of 23-Nov-2023. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1776 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
 
Theoremnf5ri 2185 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 15-Mar-2023.)
𝑥𝜑       (𝜑 → ∀𝑥𝜑)
 
Theoremnf5riOLD 2186 Obsolete proof of nf5ri 2185 as of 15-Mar-2023. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑       (𝜑 → ∀𝑥𝜑)
 
Theoremnf5rd 2187 Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → (𝜓 → ∀𝑥𝜓))
 
Theoremspimedv 2188* Version of spimed 2400 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 31-May-2019.)
(𝜒 → Ⅎ𝑥𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜒 → (𝜑 → ∃𝑥𝜓))
 
Theoremspimefv 2189* Version of spime 2401 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 31-May-2019.)
𝑥𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)
 
Theoremnfim1 2190 A closed form of nfim 1888. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1776 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       𝑥(𝜑𝜓)
 
Theoremnfan1 2191 A closed form of nfan 1891. (Contributed by Mario Carneiro, 3-Oct-2016.) df-nf 1776 changed. (Revised by Wolf Lammen, 18-Sep-2021.) (Proof shortened by Wolf Lammen, 7-Jul-2022.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       𝑥(𝜑𝜓)
 
Theorem19.3t 2192 Closed form of 19.3 2193 and version of 19.9t 2195 with a universal quantifier. (Contributed by NM, 9-Nov-2020.) (Proof shortened by BJ, 9-Oct-2022.)
(Ⅎ𝑥𝜑 → (∀𝑥𝜑𝜑))
 
Theorem19.3 2193 A wff may be quantified with a variable not free in it. Version of 19.9 2196 with a universal quantifier. Theorem 19.3 of [Margaris] p. 89. See 19.3v 1977 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑       (∀𝑥𝜑𝜑)
 
Theorem19.9d 2194 A deduction version of one direction of 19.9 2196. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) df-nf 1776 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by Wolf Lammen, 8-Jul-2022.)
(𝜓 → Ⅎ𝑥𝜑)       (𝜓 → (∃𝑥𝜑𝜑))
 
Theorem19.9t 2195 Closed form of 19.9 2196 and version of 19.3t 2192 with an existential quantifier. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 14-Jul-2020.)
(Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
 
Theorem19.9 2196 A wff may be existentially quantified with a variable not free in it. Version of 19.3 2193 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1979 for a version requiring fewer axioms. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.)
𝑥𝜑       (∃𝑥𝜑𝜑)
 
Theorem19.21t 2197 Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2198. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1776 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 3-Nov-2021.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
 
Theorem19.21 2198 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See 19.21v 1931 for a version requiring fewer axioms. See also 19.21h 2287. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1776 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑       (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
 
Theoremstdpc5 2199 An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis 𝑥𝜑 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 2011. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. See stdpc5v 1930 for a version requiring fewer axioms. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2136. (Revised by Wolf Lammen, 4-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))
 
Theorem19.21-2 2200 Version of 19.21 2198 with two quantifiers. (Contributed by NM, 4-Feb-2005.)
𝑥𝜑    &   𝑦𝜑       (∀𝑥𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝑦𝜓))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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