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Theorem List for Metamath Proof Explorer - 2101-2200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsbrimvlem 2101* Common proof template for sbrimvw 2102 and sbrimv 2315. The hypothesis is an instance of 19.21 2208. (Contributed by Wolf Lammen, 29-Jan-2024.)
(∀𝑥(𝜑 → (𝑥 = 𝑦𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓)))       ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
 
Theoremsbrimvw 2102* Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim 2314 and sbrimv 2315 based on fewer axioms, but with more disjoint variable conditions. Based on an idea of Gino Giotto. (Contributed by Wolf Lammen, 29-Jan-2024.)
([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
 
Theoremsbievw 2103* Conversion of implicit substitution to explicit substitution. Version of sbie 2544 and sbiev 2331 with more disjoint variable conditions, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by BJ, 18-Jul-2023.)
(𝑥 = 𝑦 → (𝜑𝜓))       ([𝑦 / 𝑥]𝜑𝜓)
 
Theoremsbiedvw 2104* Conversion of implicit substitution to explicit substitution (deduction version of sbievw 2103). Version of sbied 2545 and sbiedv 2546 with more disjoint variable conditions, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Gino Giotto, 29-Jan-2024.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
 
Theorem2sbievw 2105* Conversion of double implicit substitution to explicit substitution. Version of 2sbiev 2547 with more disjoint variable conditions, requiring fewer axioms. (Contributed by AV, 29-Jul-2023.) (Revised by Gino Giotto, 10-Jan-2024.)
((𝑥 = 𝑡𝑦 = 𝑢) → (𝜑𝜓))       ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑𝜓)
 
Theoremsbcom3vv 2106* Substituting 𝑦 for 𝑥 and then 𝑧 for 𝑦 is equivalent to substituting 𝑧 for both 𝑥 and 𝑦. Version of sbcom3 2548 with a disjoint variable condition using fewer axioms. (Contributed by NM, 27-May-1997.) (Revised by Giovanni Mascellani, 8-Apr-2018.) (Revised by BJ, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 19-Jan-2023.)
([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑)
 
Theoremsbievw2 2107* sbievw 2103 applied twice, avoiding a DV condition on 𝑥, 𝑦. Based on proofs by Wolf Lammen. (Contributed by Steven Nguyen, 29-Jul-2023.)
(𝑥 = 𝑤 → (𝜑𝜒))    &   (𝑤 = 𝑦 → (𝜒𝜓))       ([𝑦 / 𝑥]𝜑𝜓)
 
Theoremsbco2vv 2108* A composition law for substitution. Version of sbco2 2553 with disjoint variable conditions and fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by BJ, 22-Dec-2020.) (Proof shortened by Wolf Lammen, 29-Apr-2023.)
([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
 
Theoremequsb3 2109* Substitution in an equality. (Contributed by Raph Levien and FL, 4-Dec-2005.) Reduce axiom usage. (Revised by Wolf Lammen, 23-Jul-2023.)
([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
 
Theoremequsb3r 2110* Substitution applied to the atomic wff with equality. Variant of equsb3 2109. (Contributed by AV, 29-Jul-2023.) (Proof shortened by Wolf Lammen, 2-Sep-2023.)
([𝑦 / 𝑥]𝑧 = 𝑥𝑧 = 𝑦)
 
Theoremequsb3rOLD 2111* Obsolete version of equsb3r 2110 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 2112* Substitution applied to an atomic wff. Version of equsb1 2530 with a disjoint variable condition, which neither requires ax-12 2178 nor ax-13 2391. (Contributed by NM, 10-May-1993.) (Revised 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 2070. (Revised by Steven Nguyen, 11-Jul-2023.) (Proof shortened by Steven Nguyen, 22-Jul-2023.)
[𝑦 / 𝑥]𝑥 = 𝑦
 
Theoremequsb1vOLD 2113* Obsolete version of equsb1v 2112 as of 22-Jul-2023. Version of equsb1 2530 with a disjoint variable condition, which neither requires ax-12 2178 nor ax-13 2391. (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 2070. (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 2114 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 2115 of predicate calculus in terms of the wcel 2114 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 2801 for more information on the set theory usage of wcel 2114.)

wff 𝐴𝐵
 
Theoremwel 2115 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 2115 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 2114. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 2115 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 2114. 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 2116 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 2120 that this axiom can be recovered from its weakened version ax8v 2117 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-8 2116 should be ax8v 2117. See the comment of ax8v 2117 for more details on these matters. (Contributed by NM, 30-Jun-1993.) (Revised by BJ, 7-Dec-2020.) Use ax8 2120 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 
Theoremax8v 2117* Weakened version of ax-8 2116, with a disjoint variable condition on 𝑥, 𝑦. This should be the only proof referencing ax-8 2116, and it should be referenced only by its two weakened versions ax8v1 2118 and ax8v2 2119, from which ax-8 2116 is then rederived as ax8 2120, which shows that either ax8v 2117 or the conjunction of ax8v1 2118 and ax8v2 2119 is sufficient. (Contributed by BJ, 7-Dec-2020.) Use ax8 2120 instead. (New usage is discouraged.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 
Theoremax8v1 2118* First of two weakened versions of ax8v 2117, with an extra disjoint variable condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 
Theoremax8v2 2119* Second of two weakened versions of ax8v 2117, with an extra disjoint variable condition on 𝑦, 𝑧 see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 
Theoremax8 2120 Proof of ax-8 2116 from ax8v1 2118 and ax8v2 2119, proving sufficiency of the conjunction of the latter two weakened versions of ax8v 2117, which is itself a weakened version of ax-8 2116. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 
Theoremelequ1 2121 An identity law for the non-logical predicate. (Contributed by NM, 30-Jun-1993.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 
Theoremelsb3 2122* 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 2123* When the class variables in definition df-clel 2894 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 2115 with the class variables in wcel 2114. (Contributed by NM, 28-Jan-2004.) Revised to use equsexvw 2011 in order to remove dependencies on ax-10 2145, ax-12 2178, ax-13 2391. 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 2124 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 2128 that this axiom can be recovered from its weakened version ax9v 2125 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-9 2124 should be ax9v 2125. See the comment of ax9v 2125 for more details on these matters. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 7-Dec-2020.) Use ax9 2128 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 
Theoremax9v 2125* Weakened version of ax-9 2124, with a disjoint variable condition on 𝑥, 𝑦. This should be the only proof referencing ax-9 2124, and it should be referenced only by its two weakened versions ax9v1 2126 and ax9v2 2127, from which ax-9 2124 is then rederived as ax9 2128, which shows that either ax9v 2125 or the conjunction of ax9v1 2126 and ax9v2 2127 is sufficient. (Contributed by BJ, 7-Dec-2020.) Use ax9 2128 instead. (New usage is discouraged.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 
Theoremax9v1 2126* First of two weakened versions of ax9v 2125, with an extra disjoint variable condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 
Theoremax9v2 2127* Second of two weakened versions of ax9v 2125, with an extra disjoint variable condition on 𝑦, 𝑧 see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 
Theoremax9 2128 Proof of ax-9 2124 from ax9v1 2126 and ax9v2 2127, proving sufficiency of the conjunction of the latter two weakened versions of ax9v 2125, which is itself a weakened version of ax-9 2124. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 
Theoremelequ2 2129 An identity law for the non-logical predicate. (Contributed by NM, 21-Jun-1993.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
 
Theoremelsb4 2130* 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 2131* A form of elequ2 2129 with a universal quantifier. Its converse is the axiom of extensionality ax-ext 2794. (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 1811, ax-5 1911, ax6v 1971, ax-7 2015, ax-8 2116, and ax-9 2124, together with rule ax-gen 1797. See mmset.html#compare 1797. 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 2145, ax-11 2161, ax-12 2178, and ax-13 2391, 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 2178 has been proved to be required; see https://us.metamath.org/award2003.html#9a 2178. Metalogical independence of the other three are open problems.)

(There are additional predicate calculus axiom schemes included in set.mm such as ax-c5 36137, 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 ($d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the 𝑥 and 𝑦 in ax-6 1970 are bundled, but they are not in ax6v 1971. We also say that a scheme is bundled when it has at least one pair of bundled setvar variables. If distinct variable conditions are added to all setvar variable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax6v 1971 is the principal instance of ax-6 1970. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance ¬ ∀𝑥¬ 𝑥 = 𝑥 of ax-6 1970 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions ($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 2145, ax-11 2161, ax-12 2178, and ax-13 2391. "Translating" this to Metamath, it means that Tarski's axioms can prove any substitution instance of ax-10 2145, ax-11 2161, ax-12 2178, or ax-13 2391 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 2133, ax11w 2134, ax12w 2137, and ax13w 2140 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-10 2145, ax-11 2161, ax-12 2178, and ax-13 2391 meeting Conditions (1) and (2). (The "w" suffix stands for "weak version".) Each hypothesis of ax10w 2133, ax11w 2134, and ax12w 2137 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 2139 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 2135, ax12dgen 2138, ax13dgen1 2141, ax13dgen2 2142, ax13dgen3 2143, and ax13dgen4 2144. (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 2145, ax-11 2161, ax-12 2178, and ax-13 2391 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 1970 in an older system, so it seems the main purpose of his later ax6v 1971 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 1970 as our official axiom, we show that the degenerate instance holds in ax6dgen 2132. (Recall that in set.mm, the only statement referencing ax-6 1970 is ax6v 1971.)

The case of sp 2183 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 2041, again requiring substitution instances of 𝜑 that meet Conditions (1) and (2) above. Note that our direct proof sp 2183 requires ax-12 2178, which is not part of Tarski's system.

 
Theoremax6dgen 2132 Tarski's system uses the weaker ax6v 1971 instead of the bundled ax-6 1970, so here we show that the degenerate case of ax-6 1970 can be derived. Even though ax-6 1970 is in the list of axioms used, recall that in set.mm, the only statement referencing ax-6 1970 is ax6v 1971. We later rederive from ax6v 1971 the bundled form as ax6 2403 with the help of the auxiliary axiom schemes. (Contributed by NM, 23-Apr-2017.)
¬ ∀𝑥 ¬ 𝑥 = 𝑥
 
Theoremax10w 2133* Weak version of ax-10 2145 from which we can prove any ax-10 2145 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. It is an alias of hbn1w 2053 introduced for labeling consistency. (Contributed by NM, 9-Apr-2017.) Use hbn1w 2053 instead. (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
 
Theoremax11w 2134* Weak version of ax-11 2161 from which we can prove any ax-11 2161 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 2161, this theorem requires that 𝑥 and 𝑦 be distinct i.e. are not bundled. It is an alias of alcomiw 2050 introduced for labeling consistency. (Contributed by NM, 10-Apr-2017.) Use alcomiw 2050 instead. (New usage is discouraged.)
(𝑦 = 𝑧 → (𝜑𝜓))       (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
Theoremax11dgen 2135 Degenerate instance of ax-11 2161 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 2136* Lemma for weak version of ax-12 2178. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax12w 2137. (Contributed by NM, 10-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theoremax12w 2137* Weak version of ax-12 2178 from which we can prove any ax-12 2178 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 2139. (Contributed by NM, 10-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑦 = 𝑧 → (𝜑𝜒))       (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theoremax12dgen 2138 Degenerate instance of ax-12 2178 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 2139* Example of an application of ax12w 2137 that results in an instance of ax-12 2178 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 2043 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 2140* Weak version (principal instance) of ax-13 2391. (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 2133, ax11w 2134, and ax12w 2137. (Contributed by NM, 10-Apr-2017.)
𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
 
Theoremax13dgen1 2141 Degenerate instance of ax-13 2391 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 2142 Degenerate instance of ax-13 2391 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 2143 Degenerate instance of ax-13 2391 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 2144 Degenerate instance of ax-13 2391 where bundled variables 𝑥, 𝑦, and 𝑧 have a common substitution. Therefore, also a degenerate instance of ax13dgen1 2141, ax13dgen2 2142, and ax13dgen3 2143. 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 2145, ax-11 2161, ax-12 2178, and ax-13 2391 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 2133, ax11w 2134, ax12w 2137, and ax13w 2140, 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 2178 from all others has been shown, and independence of Tarski's ax-6 1970 from all others has been shown; see items 9a and 11 on https://us.metamath.org/award2003.html 1970.

 
1.5.1  Axiom scheme ax-10 (Quantified Negation)
 
Axiomax-10 2145 Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax10w 2133) 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 2146 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 2145 through ax-13 2391, by invoking ax10w 2133 through ax13w 2140. We encourage proving theorems *without* ax-10 2145 through ax-13 2391 and moving them up to the ax-4 1811 through ax-9 2124 section. (New usage is discouraged.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
 
Theoremhbn1 2146 Alias for ax-10 2145 to be used instead of it. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
 
Theoremhbe1 2147 The setvar 𝑥 is not free in 𝑥𝜑. Corresponds to the axiom (5) of modal logic (see also modal5 2159). (Contributed by NM, 24-Jan-1993.)
(∃𝑥𝜑 → ∀𝑥𝑥𝜑)
 
Theoremhbe1a 2148 Dual statement of hbe1 2147. Modified version of axc7e 2338 with a universally quantified consequent. (Contributed by Wolf Lammen, 15-Sep-2021.)
(∃𝑥𝑥𝜑 → ∀𝑥𝜑)
 
Theoremnf5-1 2149 One direction of nf5 2291 can be proved with a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 16-Sep-2021.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
 
Theoremnf5i 2150 Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑 → ∀𝑥𝜑)       𝑥𝜑
 
Theoremnf5dh 2151 Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 11-Oct-2021.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)
 
Theoremnf5dv 2152* Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 18-Sep-2021.) (Proof shortened by Wolf Lammen, 13-Jul-2022.)
(𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)
 
Theoremnfnaew 2153* All variables are effectively bound in a distinct variable specifier. Version of nfnae 2457 with a disjoint variable condition, which does not require ax-13 2391. (Contributed by Mario Carneiro, 11-Aug-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑧 ¬ ∀𝑥 𝑥 = 𝑦
 
Theoremnfe1 2154 The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝑥𝜑
 
Theoremnfa1 2155 The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 11-Sep-2021.) Remove dependency on ax-12 2178. (Revised by Wolf Lammen, 12-Oct-2021.)
𝑥𝑥𝜑
 
Theoremnfna1 2156 A convenience theorem particularly designed to remove dependencies on ax-11 2161 in conjunction with distinctors. (Contributed by Wolf Lammen, 2-Sep-2018.)
𝑥 ¬ ∀𝑥𝜑
 
Theoremnfia1 2157 Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥(∀𝑥𝜑 → ∀𝑥𝜓)
 
Theoremnfnf1 2158 The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2178. (Revised by Wolf Lammen, 12-Oct-2021.)
𝑥𝑥𝜑
 
Theoremmodal5 2159 The analogue in our predicate calculus of axiom (5) of modal logic S5. See also hbe1 2147. (Contributed by NM, 5-Oct-2005.)
(¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
 
Theoremnfs1v 2160* The setvar 𝑥 is not free in [𝑦 / 𝑥]𝜑 when 𝑥 and 𝑦 are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) Shorten nfs1v 2160 and hbs1 2275 combined. (Revised by Wolf Lammen, 28-Jul-2022.)
𝑥[𝑦 / 𝑥]𝜑
 
1.5.2  Axiom scheme ax-11 (Quantifier Commutation)
 
Axiomax-11 2161 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 2134) but is used as an auxiliary axiom scheme to achieve metalogical completeness. (Contributed by NM, 12-Mar-1993.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
Theoremalcoms 2162 Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.)
(∀𝑥𝑦𝜑𝜓)       (∀𝑦𝑥𝜑𝜓)
 
Theoremalcom 2163 Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 30-Jun-1993.)
(∀𝑥𝑦𝜑 ↔ ∀𝑦𝑥𝜑)
 
Theoremalrot3 2164 Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦𝑧𝜑 ↔ ∀𝑦𝑧𝑥𝜑)
 
Theoremalrot4 2165 Rotate four universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
(∀𝑥𝑦𝑧𝑤𝜑 ↔ ∀𝑧𝑤𝑥𝑦𝜑)
 
Theoremsbal 2166* 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 2167* Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)
 
Theoremsbcom2 2168* 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 2169 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-5 1911, ax-6 1970, ax-7 2015, ax-10 2145, ax-12 2178. (Revised by Wolf Lammen, 8-Jan-2018.) (Proof shortened by Wolf Lammen, 22-Aug-2020.)
(∃𝑥𝑦𝜑 ↔ ∃𝑦𝑥𝜑)
 
Theoremexcomim 2170 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 1911, ax-6 1970, ax-7 2015, ax-10 2145, ax-12 2178. (Revised by Wolf Lammen, 8-Jan-2018.)
(∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑)
 
Theoremexcom13 2171 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝜑 ↔ ∃𝑧𝑦𝑥𝜑)
 
Theoremexrot3 2172 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(∃𝑥𝑦𝑧𝜑 ↔ ∃𝑦𝑧𝑥𝜑)
 
Theoremexrot4 2173 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑧𝑤𝑥𝑦𝜑)
 
Theoremhbal 2174 If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by NM, 12-Mar-1993.)
(𝜑 → ∀𝑥𝜑)       (∀𝑦𝜑 → ∀𝑥𝑦𝜑)
 
Theoremhbald 2175 Deduction form of bound-variable hypothesis builder hbal 2174. (Contributed by NM, 2-Jan-2002.)
(𝜑 → ∀𝑦𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → (∀𝑦𝜓 → ∀𝑥𝑦𝜓))
 
Theoremhbsbw 2176* If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. Version of hbsb 2567 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 12-Aug-1993.) (Revised by Gino Giotto, 23-May-2024.)
(𝜑 → ∀𝑧𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
 
Theoremnfa2 2177 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2178. (Revised by Wolf Lammen, 18-Oct-2021.)
𝑥𝑦𝑥𝜑
 
1.5.3  Axiom scheme ax-12 (Substitution)
 
Axiomax-12 2178 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 2093). 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 36143 and was replaced with this shorter ax-12 2178 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2445. Conversely, this axiom is proved from ax-c15 36143 as theorem ax12 2446.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 36143) from the others on 19-Jan-2006. See item 9a at https://us.metamath.org/award2003.html 36143.

See ax12v 2179 and ax12v2 2180 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax12w 2137) 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 2179* This is essentially axiom ax-12 2178 weakened by additional restrictions on variables. Besides axc11r 2387, this theorem should be the only one referencing ax-12 2178 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 1911, 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 2180* It is possible to remove any restriction on 𝜑 in ax12v 2179. Same as Axiom C8 of [Monk2] p. 105. Use ax12v 2179 instead when sufficient. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-10 2145 and ax-13 2391. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theorem19.8a 2181 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 1987 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 9-Jan-1993.) Allow a shortening of sp 2183. (Revised by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
(𝜑 → ∃𝑥𝜑)
 
Theorem19.8ad 2182 If a wff is true, it is true for at least one instance. Deduction form of 19.8a 2181. (Contributed by DAW, 13-Feb-2017.)
(𝜑𝜓)       (𝜑 → ∃𝑥𝜓)
 
Theoremsp 2183 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 2073.

This theorem shows that our obsolete axiom ax-c5 36137 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 2178. It is thought the best we can do using only Tarski's axioms is spw 2041. Also see spvw 1985 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 2184 Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1797. Contrary to the rule of generalization, its closed form is valid, see sp 2183. (Contributed by NM, 5-Aug-1993.)
𝑥𝜑       𝜑
 
Theoremsps 2185 Generalization of antecedent. (Contributed by NM, 5-Jan-1993.)
(𝜑𝜓)       (∀𝑥𝜑𝜓)
 
Theorem2sp 2186 A double specialization (see sp 2183). Another double specialization, closer to PM*11.1, is 2stdpc4 2075. (Contributed by BJ, 15-Sep-2018.)
(∀𝑥𝑦𝜑𝜑)
 
Theoremspsd 2187 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))
 
Theorem19.2g 2188 Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. Use 19.2 1981 when sufficient. (Contributed by Mel L. O'Cat, 31-Mar-2008.)
(∀𝑥𝜑 → ∃𝑦𝜑)
 
Theorem19.21bi 2189 Inference form of 19.21 2208 and also deduction form of sp 2183. (Contributed by NM, 26-May-1993.)
(𝜑 → ∀𝑥𝜓)       (𝜑𝜓)
 
Theorem19.21bbi 2190 Inference removing two universal quantifiers. Version of 19.21bi 2189 with two quantifiers. (Contributed by NM, 20-Apr-1994.)
(𝜑 → ∀𝑥𝑦𝜓)       (𝜑𝜓)
 
Theorem19.23bi 2191 Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2212. (Contributed by NM, 12-Mar-1993.)
(∃𝑥𝜑𝜓)       (𝜑𝜓)
 
Theoremnexr 2192 Inference associated with the contrapositive of 19.8a 2181. (Contributed by Jeff Hankins, 26-Jul-2009.)
¬ ∃𝑥𝜑        ¬ 𝜑
 
Theoremqexmid 2193 Quantified excluded middle (see exmid 892). 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 2194 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by Wolf Lammen, 23-Nov-2023.)
(Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
 
Theoremnf5rOLD 2195 Obsolete version of nfrd 1793 as of 23-Nov-2023. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
 
Theoremnf5ri 2196 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)
 
Theoremnf5rd 2197 Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → (𝜓 → ∀𝑥𝜓))
 
Theoremspimedv 2198* Deduction version of spimev 2411. Version of spimed 2407 with a disjoint variable condition, which does not require ax-13 2391. See spime 2408 for a non-deduction version. (Contributed by NM, 14-May-1993.) (Revised by BJ, 31-May-2019.)
(𝜒 → Ⅎ𝑥𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜒 → (𝜑 → ∃𝑥𝜓))
 
Theoremspimefv 2199* Version of spime 2408 with a disjoint variable condition, which does not require ax-13 2391. (Contributed by BJ, 31-May-2019.)
𝑥𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)
 
Theoremnfim1 2200 A closed form of nfim 1897. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1786 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       𝑥(𝜑𝜓)
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45272
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