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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 2albii 1401 | Inference adding 2 universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓) | ||
Theorem | hbxfrbi 1402 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ (𝜑 → ∀𝑥𝜑) | ||
Theorem | nfbii 1403 | Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓) | ||
Theorem | nfxfr 1404 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥𝜑 | ||
Theorem | nfxfrd 1405 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜒 → Ⅎ𝑥𝜑) | ||
Theorem | alcoms 1406 | Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.) |
⊢ (∀𝑥∀𝑦𝜑 → 𝜓) ⇒ ⊢ (∀𝑦∀𝑥𝜑 → 𝜓) | ||
Theorem | hbal 1407 | If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑) | ||
Theorem | alcom 1408 | Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑦∀𝑥𝜑) | ||
Theorem | alrimdh 1409 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒)) | ||
Theorem | albidh 1410 | Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) | ||
Theorem | 19.26 1411 | Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) |
⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓)) | ||
Theorem | 19.26-2 1412 | Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.) |
⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓)) | ||
Theorem | 19.26-3an 1413 | Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.) |
⊢ (∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓 ∧ ∀𝑥𝜒)) | ||
Theorem | 19.33 1414 | Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | alrot3 1415 | Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∀𝑥∀𝑦∀𝑧𝜑 ↔ ∀𝑦∀𝑧∀𝑥𝜑) | ||
Theorem | alrot4 1416 | Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 28-Jun-2014.) |
⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑) | ||
Theorem | albiim 1417 | Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑))) | ||
Theorem | 2albiim 1418 | Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.) |
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) ↔ (∀𝑥∀𝑦(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦(𝜓 → 𝜑))) | ||
Theorem | hband 1419 | Deduction form of bound-variable hypothesis builder hban 1480. (Contributed by NM, 2-Jan-2002.) |
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜒))) | ||
Theorem | hb3and 1420 | Deduction form of bound-variable hypothesis builder hb3an 1483. (Contributed by NM, 17-Feb-2013.) |
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) & ⊢ (𝜑 → (𝜃 → ∀𝑥𝜃)) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → ∀𝑥(𝜓 ∧ 𝜒 ∧ 𝜃))) | ||
Theorem | hbald 1421 | Deduction form of bound-variable hypothesis builder hbal 1407. (Contributed by NM, 2-Jan-2002.) |
⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓)) | ||
Syntax | wex 1422 | Extend wff definition to include the existential quantifier ("there exists"). |
wff ∃𝑥𝜑 | ||
Axiom | ax-ie1 1423 | 𝑥 is bound in ∃𝑥𝜑. One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | ||
Axiom | ax-ie2 1424 | Define existential quantification. ∃𝑥𝜑 means "there exists at least one set 𝑥 such that 𝜑 is true." One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) | ||
Theorem | hbe1 1425 | 𝑥 is not free in ∃𝑥𝜑. (Contributed by NM, 5-Aug-1993.) |
⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | ||
Theorem | nfe1 1426 | 𝑥 is not free in ∃𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥∃𝑥𝜑 | ||
Theorem | 19.23ht 1427 | Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 1-Feb-2015.) |
⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) | ||
Theorem | 19.23h 1428 | Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.) |
⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)) | ||
Theorem | alnex 1429 | Theorem 19.7 of [Margaris] p. 89. To read this intuitionistically, think of it as "if 𝜑 can be refuted for all 𝑥, then it is not possible to find an 𝑥 for which 𝜑 holds" (and likewise for the converse). Comparing this with dfexdc 1431 illustrates that statements which look similar (to someone used to classical logic) can be different intuitionistically due to different placement of negations. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 1-Feb-2015.) (Revised by Mario Carneiro, 12-May-2015.) |
⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | ||
Theorem | nex 1430 | Generalization rule for negated wff. (Contributed by NM, 18-May-1994.) |
⊢ ¬ 𝜑 ⇒ ⊢ ¬ ∃𝑥𝜑 | ||
Theorem | dfexdc 1431 | Defining ∃𝑥𝜑 given decidability. It is common in classical logic to define ∃𝑥𝜑 as ¬ ∀𝑥¬ 𝜑 but in intuitionistic logic without a decidability condition, that is only an implication not an equivalence, as seen at exalim 1432. (Contributed by Jim Kingdon, 15-Mar-2018.) |
⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)) | ||
Theorem | exalim 1432 | One direction of a classical definition of existential quantification. One direction of Definition of [Margaris] p. 49. For a decidable proposition, this is an equivalence, as seen as dfexdc 1431. (Contributed by Jim Kingdon, 29-Jul-2018.) |
⊢ (∃𝑥𝜑 → ¬ ∀𝑥 ¬ 𝜑) | ||
The equality predicate was introduced above in wceq 1285 for use by df-tru 1288. See the comments in that section. In this section, we continue with the first "real" use of it. | ||
Theorem | weq 1433 |
Extend wff definition to include atomic formulas using the equality
predicate.
(Instead of introducing weq 1433 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 wceq 1285. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1433 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1285. Note: To see the proof steps of this syntax proof, type "show proof weq /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.) |
wff 𝑥 = 𝑦 | ||
Syntax | wcel 1434 |
Extend wff definition to include the membership connective between
classes.
(The purpose of introducing wff 𝐴 ∈ 𝐵 here is to allow us to express i.e. "prove" the wel 1435 of predicate calculus in terms of the wceq 1285 of set theory, so that we don't "overload" the ∈ connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables 𝐴 and 𝐵 are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus.) |
wff 𝐴 ∈ 𝐵 | ||
Theorem | wel 1435 |
Extend wff definition to include atomic formulas with the epsilon
(membership) predicate. This is read "𝑥 is an element of
𝑦," "𝑥 is a member of 𝑦,"
"𝑥 belongs to 𝑦,"
or "𝑦 contains 𝑥." Note: The
phrase "𝑦 includes
𝑥 " means "𝑥 is a
subset of 𝑦;" to use it also for
𝑥
∈ 𝑦, as some
authors occasionally do, is poor form and causes
confusion, according to George Boolos (1992 lecture at MIT).
This syntactical construction introduces a binary non-logical predicate symbol ∈ (epsilon) into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for ∈ apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments. (Instead of introducing wel 1435 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 1434. This lets us avoid overloading the ∈ connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1435 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1434. Note: To see the proof steps of this syntax proof, type "show proof wel /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.) |
wff 𝑥 ∈ 𝑦 | ||
Axiom | ax-8 1436 |
Axiom of Equality. One of the equality and substitution axioms of
predicate calculus with equality. This is similar to, but not quite, a
transitive law for equality (proved later as equtr 1637). Axiom scheme C8'
in [Megill] p. 448 (p. 16 of the preprint).
Also appears as Axiom C7 of
[Monk2] p. 105.
Axioms ax-8 1436 through ax-16 1737 are the axioms having to do with equality, substitution, and logical properties of our binary predicate ∈ (which later in set theory will mean "is a member of"). Note that all axioms except ax-16 1737 and ax-17 1460 are still valid even when 𝑥, 𝑦, and 𝑧 are replaced with the same variable because they do not have any distinct variable (Metamath's $d) restrictions. Distinct variable restrictions are required for ax-16 1737 and ax-17 1460 only. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) | ||
Axiom | ax-10 1437 |
Axiom of Quantifier Substitution. One of the equality and substitution
axioms of predicate calculus with equality. Appears as Lemma L12 in
[Megill] p. 445 (p. 12 of the preprint).
The original version of this axiom was ax-10o 1646 ("o" for "old") and was replaced with this shorter ax-10 1437 in May 2008. The old axiom is proved from this one as theorem ax10o 1645. Conversely, this axiom is proved from ax-10o 1646 as theorem ax10 1647. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
Axiom | ax-11 1438 |
Axiom of Variable 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 1809). 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.
Variants of this axiom which are equivalent in classical logic but which have not been shown to be equivalent for intuitionistic logic are ax11v 1750, ax11v2 1743 and ax-11o 1746. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Axiom | ax-i12 1439 |
Axiom of Quantifier Introduction. One of the equality and substitution
axioms of predicate calculus with equality. Informally, it says that
whenever 𝑧 is distinct from 𝑥 and
𝑦,
and 𝑥 =
𝑦 is true,
then 𝑥 = 𝑦 quantified with 𝑧 is also
true. In other words, 𝑧
is irrelevant to the truth of 𝑥 = 𝑦. Axiom scheme C9' in [Megill]
p. 448 (p. 16 of the preprint). It apparently does not otherwise appear
in the literature but is easily proved from textbook predicate calculus by
cases.
This axiom has been modified from the original ax-12 1443 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) | ||
Axiom | ax-bndl 1440 |
Axiom of bundling. The general idea of this axiom is that two variables
are either distinct or non-distinct. That idea could be expressed as
∀𝑧𝑧 = 𝑥 ∨ ¬ ∀𝑧𝑧 = 𝑥. However, we instead choose an axiom
which has many of the same consequences, but which is different with
respect to a universe which contains only one object. ∀𝑧𝑧 = 𝑥 holds
if 𝑧 and 𝑥 are the same variable,
likewise for 𝑧 and 𝑦,
and ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧𝑥 = 𝑦) holds if 𝑧 is distinct from
the others (and the universe has at least two objects).
As with other statements of the form "x is decidable (either true or false)", this does not entail the full Law of the Excluded Middle (which is the proposition that all statements are decidable), but instead merely the assertion that particular kinds of statements are decidable (or in this case, an assertion similar to decidability). This axiom implies ax-i12 1439 as can be seen at axi12 1448. Whether ax-bndl 1440 can be proved from the remaining axioms including ax-i12 1439 is not known. The reason we call this "bundling" is that a statement without a distinct variable constraint "bundles" together two statements, one in which the two variables are the same and one in which they are different. (Contributed by Mario Carneiro and Jim Kingdon, 14-Mar-2018.) |
⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) | ||
Axiom | ax-4 1441 |
Axiom of Specialization. A quantified wff implies the wff without a
quantifier (i.e. an instance, or special case, of the generalized wff).
In other words if something is true for all 𝑥, it is true for any
specific 𝑥 (that would typically occur as a free
variable in the wff
substituted for 𝜑). (A free variable is one that does
not occur in
the scope of a quantifier: 𝑥 and 𝑦 are both free in 𝑥 = 𝑦,
but only 𝑥 is free in ∀𝑦𝑥 = 𝑦.) Axiom scheme C5' in [Megill]
p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski]
p. 67 (under his system S2, defined in the last paragraph on p. 77).
Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1379. Conditional forms of the converse are given by ax-12 1443, ax-16 1737, and ax-17 1460. Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from 𝑥 for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 1700. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | sp 1442 | Specialization. Another name for ax-4 1441. (Contributed by NM, 21-May-2008.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | ax-12 1443 | Rederive the original version of the axiom from ax-i12 1439. (Contributed by Mario Carneiro, 3-Feb-2015.) |
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) | ||
Theorem | ax12or 1444 | Another name for ax-i12 1439. (Contributed by NM, 3-Feb-2015.) |
⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) | ||
Axiom | ax-13 1445 | Axiom of Equality. 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 the ∈ binary predicate. Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | ||
Axiom | ax-14 1446 | Axiom of Equality. 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 the ∈ binary predicate. Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | ||
Theorem | hbequid 1447 | Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (The proof uses only ax-5 1377, ax-8 1436, ax-12 1443, and ax-gen 1379. This shows that this can be proved without ax-9 1465, even though the theorem equid 1630 cannot be. A shorter proof using ax-9 1465 is obtainable from equid 1630 and hbth 1393.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) |
⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥) | ||
Theorem | axi12 1448 | Proof that ax-i12 1439 follows from ax-bndl 1440. So that we can track which theorems rely on ax-bndl 1440, proofs should reference ax-i12 1439 rather than this theorem. (Contributed by Jim Kingdon, 17-Aug-2018.) (New usage is discouraged). (Proof modification is discouraged.) |
⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) | ||
Theorem | alequcom 1449 | Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
Theorem | alequcoms 1450 | A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑) | ||
Theorem | nalequcoms 1451 | A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 2-Feb-2015.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑) | ||
Theorem | nfr 1452 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) |
⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | ||
Theorem | nfri 1453 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝜑 → ∀𝑥𝜑) | ||
Theorem | nfrd 1454 | Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | ||
Theorem | alimd 1455 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) | ||
Theorem | alrimi 1456 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → ∀𝑥𝜓) | ||
Theorem | nfd 1457 | Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝜓) | ||
Theorem | nfdh 1458 | Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝜓) | ||
Theorem | nfrimi 1459 | Moving an antecedent outside Ⅎ. (Contributed by Jim Kingdon, 23-Mar-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥(𝜑 → 𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝜓) | ||
Axiom | ax-17 1460* |
Axiom to quantify a variable over a formula in which it does not occur.
Axiom C5 in [Megill] p. 444 (p. 11 of the
preprint). Also appears as
Axiom B6 (p. 75) of system S2 of [Tarski]
p. 77 and Axiom C5-1 of
[Monk2] p. 113.
(Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) | ||
Theorem | a17d 1461* | ax-17 1460 with antecedent. (Contributed by NM, 1-Mar-2013.) |
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | ||
Theorem | nfv 1462* | If 𝑥 is not present in 𝜑, then 𝑥 is not free in 𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 | ||
Theorem | nfvd 1463* | nfv 1462 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1518. (Contributed by Mario Carneiro, 6-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) | ||
Axiom | ax-i9 1464 | Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. One thing this axiom tells us is that at least one thing exists (although ax-4 1441 and possibly others also tell us that, i.e. they are not valid in the empty domain of a "free logic"). In this form (not requiring that 𝑥 and 𝑦 be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) Another name for this theorem is a9e 1627, which has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.) |
⊢ ∃𝑥 𝑥 = 𝑦 | ||
Theorem | ax-9 1465 | Derive ax-9 1465 from ax-i9 1464, the modified version for intuitionistic logic. Although ax-9 1465 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1464. (Contributed by NM, 3-Feb-2015.) |
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | ||
Theorem | equidqe 1466 | equid 1630 with some quantification and negation without using ax-4 1441 or ax-17 1460. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) |
⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥 | ||
Theorem | ax4sp1 1467 | A special case of ax-4 1441 without using ax-4 1441 or ax-17 1460. (Contributed by NM, 13-Jan-2011.) |
⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥) | ||
Axiom | ax-ial 1468 | 𝑥 is not free in ∀𝑥𝜑. One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | ||
Axiom | ax-i5r 1469 | Axiom of quantifier collection. (Contributed by Mario Carneiro, 31-Jan-2015.) |
⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓)) | ||
Theorem | spi 1470 | Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.) |
⊢ ∀𝑥𝜑 ⇒ ⊢ 𝜑 | ||
Theorem | sps 1471 | Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | spsd 1472 | Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) | ||
Theorem | nfbidf 1473 | An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒)) | ||
Theorem | hba1 1474 | 𝑥 is not free in ∀𝑥𝜑. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | ||
Theorem | nfa1 1475 | 𝑥 is not free in ∀𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥∀𝑥𝜑 | ||
Theorem | a5i 1476 | Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑥𝜓) | ||
Theorem | nfnf1 1477 | 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥Ⅎ𝑥𝜑 | ||
Theorem | hbim 1478 | If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 → 𝜓). (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (Revised by Mario Carneiro, 2-Feb-2015.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) | ||
Theorem | hbor 1479 | If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∨ 𝜓). (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ ((𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | hban 1480 | If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | hbbi 1481 | If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ↔ 𝜓). (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ ((𝜑 ↔ 𝜓) → ∀𝑥(𝜑 ↔ 𝜓)) | ||
Theorem | hb3or 1482 | If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑 ∨ 𝜓 ∨ 𝜒). (Contributed by NM, 14-Sep-2003.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝜒 → ∀𝑥𝜒) ⇒ ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) → ∀𝑥(𝜑 ∨ 𝜓 ∨ 𝜒)) | ||
Theorem | hb3an 1483 | If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑 ∧ 𝜓 ∧ 𝜒). (Contributed by NM, 14-Sep-2003.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝜒 → ∀𝑥𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)) | ||
Theorem | hba2 1484 | Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) |
⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦∀𝑥𝜑) | ||
Theorem | hbia1 1485 | Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) |
⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → ∀𝑥𝜓)) | ||
Theorem | 19.3h 1486 | A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-May-2007.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∀𝑥𝜑 ↔ 𝜑) | ||
Theorem | 19.3 1487 | A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥𝜑 ↔ 𝜑) | ||
Theorem | 19.16 1488 | Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∀𝑥𝜓)) | ||
Theorem | 19.17 1489 | Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ 𝜓)) | ||
Theorem | 19.21h 1490 | Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." New proofs should use 19.21 1516 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) | ||
Theorem | 19.21bi 1491 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → ∀𝑥𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | 19.21bbi 1492 | Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.) |
⊢ (𝜑 → ∀𝑥∀𝑦𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | 19.27h 1493 | Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓)) | ||
Theorem | 19.27 1494 | Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓)) | ||
Theorem | 19.28h 1495 | Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) | ||
Theorem | 19.28 1496 | Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) | ||
Theorem | nfan1 1497 | A closed form of nfan 1498. (Contributed by Mario Carneiro, 3-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) | ||
Theorem | nfan 1498 | If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) | ||
Theorem | nf3an 1499 | If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑 ∧ 𝜓 ∧ 𝜒). (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 ⇒ ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓 ∧ 𝜒) | ||
Theorem | nford 1500 | If in a context 𝑥 is not free in 𝜓 and 𝜒, it is not free in (𝜓 ∨ 𝜒). (Contributed by Jim Kingdon, 29-Oct-2019.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∨ 𝜒)) |
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