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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sbrimvlem 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 = 𝑦 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓))) ⇒ ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) | ||
Theorem | sbrimvw 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](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) | ||
Theorem | sbievw 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 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | ||
Theorem | sbiedvw 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.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) | ||
Theorem | 2sbievw 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 = 𝑡 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ 𝜓) | ||
Theorem | sbcom3vv 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) | ||
Theorem | sbievw2 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 → (𝜑 ↔ 𝜒)) & ⊢ (𝑤 = 𝑦 → (𝜒 ↔ 𝜓)) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | ||
Theorem | sbco2vv 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) | ||
Theorem | equsb3 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 = 𝑧 ↔ 𝑦 = 𝑧) | ||
Theorem | equsb3r 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 = 𝑥 ↔ 𝑧 = 𝑦) | ||
Theorem | equsb3rOLD 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 = 𝑥 ↔ 𝑧 = 𝑦) | ||
Theorem | equsb1v 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 | ||
Theorem | equsb1vOLD 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 | ||
Syntax | wcel 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 𝐴 ∈ 𝐵 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 𝐴 ∈ 𝐵 | ||
Theorem | wel 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 𝑥 ∈ 𝑦, 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 𝑥 ∈ 𝑦 | ||
Axiom | ax-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 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | ||
Theorem | ax8v 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 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | ||
Theorem | ax8v1 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 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | ||
Theorem | ax8v2 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 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | ||
Theorem | ax8 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 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | ||
Theorem | elequ1 2121 | An identity law for the non-logical predicate. (Contributed by NM, 30-Jun-1993.) |
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) | ||
Theorem | elsb3 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]𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧) | ||
Theorem | cleljust 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.) |
⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦)) | ||
Axiom | ax-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 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | ||
Theorem | ax9v 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 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | ||
Theorem | ax9v1 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 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | ||
Theorem | ax9v2 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 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | ||
Theorem | ax9 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 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | ||
Theorem | elequ2 2129 | An identity law for the non-logical predicate. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | ||
Theorem | elsb4 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]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) | ||
Theorem | elequ2g 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(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | ||
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 → (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 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 ∀𝑥𝜑 → 𝜑 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. | ||
Theorem | ax6dgen 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 | ||
Theorem | ax10w 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 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | ||
Theorem | ax11w 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 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | ||
Theorem | ax11dgen 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.) |
⊢ (∀𝑥∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | ||
Theorem | ax12wlem 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 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | ax12w 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | ax12dgen 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 = 𝑥 → 𝜑))) | ||
Theorem | ax12wdemo 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, (𝑥 ∈ 𝑦 ∧ ∀𝑥𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧𝑦 ∈ 𝑥), 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(𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧 𝑦 ∈ 𝑥) → ∀𝑥(𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧 𝑦 ∈ 𝑥)))) | ||
Theorem | ax13w 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)) | ||
Theorem | ax13dgen1 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)) | ||
Theorem | ax13dgen2 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)) | ||
Theorem | ax13dgen3 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)) | ||
Theorem | ax13dgen4 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)) | ||
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. | ||
Axiom | ax-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) | ||
Theorem | hbn1 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) | ||
Theorem | hbe1 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 → ∀𝑥∃𝑥𝜑) | ||
Theorem | hbe1a 2148 | Dual statement of hbe1 2147. Modified version of axc7e 2338 with a universally quantified consequent. (Contributed by Wolf Lammen, 15-Sep-2021.) |
⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) | ||
Theorem | nf5-1 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) | ||
Theorem | nf5i 2150 | Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ Ⅎ𝑥𝜑 | ||
Theorem | nf5dh 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) | ||
Theorem | nf5dv 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) | ||
Theorem | nfnaew 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.) |
⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦 | ||
Theorem | nfe1 2154 | The setvar 𝑥 is not free in ∃𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥∃𝑥𝜑 | ||
Theorem | nfa1 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.) |
⊢ Ⅎ𝑥∀𝑥𝜑 | ||
Theorem | nfna1 2156 | A convenience theorem particularly designed to remove dependencies on ax-11 2161 in conjunction with distinctors. (Contributed by Wolf Lammen, 2-Sep-2018.) |
⊢ Ⅎ𝑥 ¬ ∀𝑥𝜑 | ||
Theorem | nfia1 2157 | Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥(∀𝑥𝜑 → ∀𝑥𝜓) | ||
Theorem | nfnf1 2158 | The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2178. (Revised by Wolf Lammen, 12-Oct-2021.) |
⊢ Ⅎ𝑥Ⅎ𝑥𝜑 | ||
Theorem | modal5 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) | ||
Theorem | nfs1v 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.) |
⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | ||
Axiom | ax-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.) |
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | ||
Theorem | alcoms 2162 | Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.) |
⊢ (∀𝑥∀𝑦𝜑 → 𝜓) ⇒ ⊢ (∀𝑦∀𝑥𝜑 → 𝜓) | ||
Theorem | alcom 2163 | Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 30-Jun-1993.) |
⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑦∀𝑥𝜑) | ||
Theorem | alrot3 2164 | Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∀𝑥∀𝑦∀𝑧𝜑 ↔ ∀𝑦∀𝑧∀𝑥𝜑) | ||
Theorem | alrot4 2165 | Rotate four 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 | sbal 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) | ||
Theorem | sbalv 2167* | Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) ⇒ ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓) | ||
Theorem | sbcom2 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) | ||
Theorem | excom 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.) |
⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑) | ||
Theorem | excomim 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.) |
⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑) | ||
Theorem | excom13 2171 | Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.) |
⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑧∃𝑦∃𝑥𝜑) | ||
Theorem | exrot3 2172 | Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.) |
⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑦∃𝑧∃𝑥𝜑) | ||
Theorem | exrot4 2173 | Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.) |
⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑧∃𝑤∃𝑥∃𝑦𝜑) | ||
Theorem | hbal 2174 | If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by NM, 12-Mar-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑) | ||
Theorem | hbald 2175 | Deduction form of bound-variable hypothesis builder hbal 2174. (Contributed by NM, 2-Jan-2002.) |
⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓)) | ||
Theorem | hbsbw 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) | ||
Theorem | nfa2 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.) |
⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑 | ||
Axiom | ax-12 2178 |
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 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 = 𝑦 → 𝜑))) | ||
Theorem | ax12v 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 = 𝑦 → 𝜑))) | ||
Theorem | ax12v2 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 = 𝑦 → 𝜑))) | ||
Theorem | 19.8a 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) | ||
Theorem | 19.8ad 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.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | sp 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.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | spi 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 | ||
Theorem | sps 2185 | Generalization of antecedent. (Contributed by NM, 5-Jan-1993.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | 2sp 2186 | A double specialization (see sp 2183). Another double specialization, closer to PM*11.1, is 2stdpc4 2075. (Contributed by BJ, 15-Sep-2018.) |
⊢ (∀𝑥∀𝑦𝜑 → 𝜑) | ||
Theorem | spsd 2187 | Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) | ||
Theorem | 19.2g 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) | ||
Theorem | 19.21bi 2189 | Inference form of 19.21 2208 and also deduction form of sp 2183. (Contributed by NM, 26-May-1993.) |
⊢ (𝜑 → ∀𝑥𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | 19.21bbi 2190 | Inference removing two universal quantifiers. Version of 19.21bi 2189 with two quantifiers. (Contributed by NM, 20-Apr-1994.) |
⊢ (𝜑 → ∀𝑥∀𝑦𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | 19.23bi 2191 | Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2212. (Contributed by NM, 12-Mar-1993.) |
⊢ (∃𝑥𝜑 → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | nexr 2192 | Inference associated with the contrapositive of 19.8a 2181. (Contributed by Jeff Hankins, 26-Jul-2009.) |
⊢ ¬ ∃𝑥𝜑 ⇒ ⊢ ¬ 𝜑 | ||
Theorem | qexmid 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) | ||
Theorem | nf5r 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)) | ||
Theorem | nf5rOLD 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)) | ||
Theorem | nf5ri 2196 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 15-Mar-2023.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝜑 → ∀𝑥𝜑) | ||
Theorem | nf5rd 2197 | Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | ||
Theorem | spimedv 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 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓)) | ||
Theorem | spimefv 2199* | Version of spime 2408 with a disjoint variable condition, which does not require ax-13 2391. (Contributed by BJ, 31-May-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | nfim1 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.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ Ⅎ𝑥(𝜑 → 𝜓) |
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