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Theorem List for Intuitionistic Logic Explorer - 1401-1500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnfe1 1401 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝑥𝜑

Theorem19.23ht 1402 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(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))

Theorem19.23h 1403 Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.)
(𝜓 → ∀𝑥𝜓)       (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theoremalnex 1404 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 1406 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)

Theoremnex 1405 Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
¬ 𝜑        ¬ ∃𝑥𝜑

Theoremdfexdc 1406 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 1407. (Contributed by Jim Kingdon, 15-Mar-2018.)
(DECID𝑥𝜑 → (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑))

Theoremexalim 1407 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 1406. (Contributed by Jim Kingdon, 29-Jul-2018.)
(∃𝑥𝜑 → ¬ ∀𝑥 ¬ 𝜑)

1.3.2  Equality predicate (continued)

The equality predicate was introduced above in wceq 1259 for use by df-tru 1262. See the comments in that section. In this section, we continue with the first "real" use of it.

Theoremweq 1408 Extend wff definition to include atomic formulas using the equality predicate.

(Instead of introducing weq 1408 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 1259. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1408 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1259. 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

Syntaxwcel 1409 Extend wff definition to include the membership connective between classes.

(The purpose of introducing wff 𝐴𝐵 here is to allow us to express i.e. "prove" the wel 1410 of predicate calculus in terms of the wceq 1259 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 𝐴𝐵

Theoremwel 1410 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 𝑥𝑦, 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 1410 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 1409. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1410 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1409. 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 𝑥𝑦

Axiomax-8 1411 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 1611). Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105.

Axioms ax-8 1411 through ax-16 1711 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 1711 and ax-17 1435 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 1711 and ax-17 1435 only. (Contributed by NM, 5-Aug-1993.)

(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Axiomax-10 1412 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 1620 ("o" for "old") and was replaced with this shorter ax-10 1412 in May 2008. The old axiom is proved from this one as theorem ax10o 1619. Conversely, this axiom is proved from ax-10o 1620 as theorem ax10 1621. (Contributed by NM, 5-Aug-1993.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Axiomax-11 1413 Axiom of Variable 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 1782). 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 1724, ax11v2 1717 and ax-11o 1720. (Contributed by NM, 5-Aug-1993.)

(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Axiomax-i12 1414 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 1418 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)

(∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Axiomax-bndl 1415 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 𝑥𝑧(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 1414 as can be seen at axi12 1423. Whether ax-bndl 1415 can be proved from the remaining axioms including ax-i12 1414 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 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Axiomax-4 1416 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 1354. Conditional forms of the converse are given by ax-12 1418, ax-16 1711, and ax-17 1435.

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 1674.

(Contributed by NM, 5-Aug-1993.)

(∀𝑥𝜑𝜑)

Theoremsp 1417 Specialization. Another name for ax-4 1416. (Contributed by NM, 21-May-2008.)
(∀𝑥𝜑𝜑)

Theoremax-12 1418 Rederive the original version of the axiom from ax-i12 1414. (Contributed by Mario Carneiro, 3-Feb-2015.)
(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Theoremax12or 1419 Another name for ax-i12 1414. (Contributed by NM, 3-Feb-2015.)
(∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Axiomax-13 1420 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 → (𝑥𝑧𝑦𝑧))

Axiomax-14 1421 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 → (𝑧𝑥𝑧𝑦))

Theoremhbequid 1422 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 1352, ax-8 1411, ax-12 1418, and ax-gen 1354. This shows that this can be proved without ax-9 1440, even though the theorem equid 1605 cannot be. A shorter proof using ax-9 1440 is obtainable from equid 1605 and hbth 1368.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.)
(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)

Theoremaxi12 1423 Proof that ax-i12 1414 follows from ax-bndl 1415. So that we can track which theorems rely on ax-bndl 1415, proofs should reference ax-i12 1414 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)))

Theoremalequcom 1424 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)

Theoremalequcoms 1425 A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)
(∀𝑥 𝑥 = 𝑦𝜑)       (∀𝑦 𝑦 = 𝑥𝜑)

Theoremnalequcoms 1426 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 2-Feb-2015.)
(¬ ∀𝑥 𝑥 = 𝑦𝜑)       (¬ ∀𝑦 𝑦 = 𝑥𝜑)

Theoremnfr 1427 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)
(Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Theoremnfri 1428 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       (𝜑 → ∀𝑥𝜑)

Theoremnfrd 1429 Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → (𝜓 → ∀𝑥𝜓))

Theoremalimd 1430 Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Theoremalrimi 1431 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑𝜓)       (𝜑 → ∀𝑥𝜓)

Theoremnfd 1432 Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

Theoremnfdh 1433 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)

Theoremnfrimi 1434 Moving an antecedent outside . (Contributed by Jim Kingdon, 23-Mar-2018.)
𝑥𝜑    &   𝑥(𝜑𝜓)       (𝜑 → Ⅎ𝑥𝜓)

1.3.3  Axiom ax-17 - first use of the \$d distinct variable statement

Axiomax-17 1435* 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)

Theorema17d 1436* ax-17 1435 with antecedent. (Contributed by NM, 1-Mar-2013.)
(𝜑 → (𝜓 → ∀𝑥𝜓))

Theoremnfv 1437* If 𝑥 is not present in 𝜑, then 𝑥 is not free in 𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑

Theoremnfvd 1438* nfv 1437 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1493. (Contributed by Mario Carneiro, 6-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)

1.3.4  Introduce Axiom of Existence

Axiomax-i9 1439 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 1416 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 1602, which has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.)
𝑥 𝑥 = 𝑦

Theoremax-9 1440 Derive ax-9 1440 from ax-i9 1439, the modified version for intuitionistic logic. Although ax-9 1440 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1439. (Contributed by NM, 3-Feb-2015.)
¬ ∀𝑥 ¬ 𝑥 = 𝑦

Theoremequidqe 1441 equid 1605 with some quantification and negation without using ax-4 1416 or ax-17 1435. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.)
¬ ∀𝑦 ¬ 𝑥 = 𝑥

Theoremax4sp1 1442 A special case of ax-4 1416 without using ax-4 1416 or ax-17 1435. (Contributed by NM, 13-Jan-2011.)
(∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)

Axiomax-ial 1443 𝑥 is not free in 𝑥𝜑. One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
(∀𝑥𝜑 → ∀𝑥𝑥𝜑)

Axiomax-i5r 1444 Axiom of quantifier collection. (Contributed by Mario Carneiro, 31-Jan-2015.)
((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑𝜓))

1.3.6  Predicate calculus including ax-4, without distinct variables

Theoremspi 1445 Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.)
𝑥𝜑       𝜑

Theoremsps 1446 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theoremspsd 1447 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))

Theoremnfbidf 1448 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))

Theoremhba1 1449 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 → ∀𝑥𝑥𝜑)

Theoremnfa1 1450 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝑥𝜑

Theorema5i 1451 Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.)
(∀𝑥𝜑𝜓)       (∀𝑥𝜑 → ∀𝑥𝜓)

Theoremnfnf1 1452 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝑥𝜑

Theoremhbim 1453 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (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)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theoremhbor 1454 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theoremhban 1455 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theoremhbbi 1456 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by NM, 5-Aug-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theoremhb3or 1457 If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑𝜓𝜒). (Contributed by NM, 14-Sep-2003.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝜒 → ∀𝑥𝜒)       ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))

Theoremhb3an 1458 If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑𝜓𝜒). (Contributed by NM, 14-Sep-2003.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝜒 → ∀𝑥𝜒)       ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))

Theoremhba2 1459 Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
(∀𝑦𝑥𝜑 → ∀𝑥𝑦𝑥𝜑)

Theoremhbia1 1460 Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → ∀𝑥𝜓))

Theorem19.3h 1461 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)       (∀𝑥𝜑𝜑)

Theorem19.3 1462 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.)
𝑥𝜑       (∀𝑥𝜑𝜑)

Theorem19.16 1463 Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 ↔ ∀𝑥𝜓))

Theorem19.17 1464 Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜓       (∀𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))

Theorem19.21h 1465 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 1491 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)       (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Theorem19.21bi 1466 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(𝜑 → ∀𝑥𝜓)       (𝜑𝜓)

Theorem19.21bbi 1467 Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.)
(𝜑 → ∀𝑥𝑦𝜓)       (𝜑𝜓)

Theorem19.27h 1468 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(𝜓 → ∀𝑥𝜓)       (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.27 1469 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
𝑥𝜓       (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.28h 1470 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(𝜑 → ∀𝑥𝜑)       (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Theorem19.28 1471 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
𝑥𝜑       (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Theoremnfan1 1472 A closed form of nfan 1473. (Contributed by Mario Carneiro, 3-Oct-2016.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       𝑥(𝜑𝜓)

Theoremnfan 1473 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)

Theoremnf3an 1474 If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑𝜓𝜒). (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑    &   𝑥𝜓    &   𝑥𝜒       𝑥(𝜑𝜓𝜒)

Theoremnford 1475 If in a context 𝑥 is not free in 𝜓 and 𝜒, it is not free in (𝜓𝜒). (Contributed by Jim Kingdon, 29-Oct-2019.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))

Theoremnfand 1476 If in a context 𝑥 is not free in 𝜓 and 𝜒, it is not free in (𝜓𝜒). (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))

Theoremnf3and 1477 Deduction form of bound-variable hypothesis builder nf3an 1474. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑥𝜃)       (𝜑 → Ⅎ𝑥(𝜓𝜒𝜃))

Theoremhbim1 1478 A closed form of hbim 1453. (Contributed by NM, 5-Aug-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theoremnfim1 1479 A closed form of nfim 1480. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       𝑥(𝜑𝜓)

Theoremnfim 1480 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)

Theoremhbimd 1481 Deduction form of bound-variable hypothesis builder hbim 1453. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))       (𝜑 → ((𝜓𝜒) → ∀𝑥(𝜓𝜒)))

Theoremnfor 1482 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by Jim Kingdon, 11-Mar-2018.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)

Theoremhbbid 1483 Deduction form of bound-variable hypothesis builder hbbi 1456. (Contributed by NM, 1-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))       (𝜑 → ((𝜓𝜒) → ∀𝑥(𝜓𝜒)))

Theoremnfal 1484 If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       𝑥𝑦𝜑

Theoremnfnf 1485 If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
𝑥𝜑       𝑥𝑦𝜑

Theoremnfalt 1486 Closed form of nfal 1484. (Contributed by Jim Kingdon, 11-May-2018.)
(∀𝑦𝑥𝜑 → Ⅎ𝑥𝑦𝜑)

Theoremnfa2 1487 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝑦𝑥𝜑

Theoremnfia1 1488 Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥(∀𝑥𝜑 → ∀𝑥𝜓)

Theorem19.21ht 1489 Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (New usage is discouraged.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Theorem19.21t 1490 Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Theorem19.21 1491 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑       (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Theoremstdpc5 1492 An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis 𝑥𝜑 can be thought of as emulating "𝑥 is not free in 𝜑." With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example 𝑥 would not (for us) be free in 𝑥 = 𝑥 by nfequid 1606. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))

Theoremnfimd 1493 If in a context 𝑥 is not free in 𝜓 and 𝜒, it is not free in (𝜓𝜒). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))

Theoremaaanh 1494 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
(𝜑 → ∀𝑦𝜑)    &   (𝜓 → ∀𝑥𝜓)       (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))

Theoremaaan 1495 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
𝑦𝜑    &   𝑥𝜓       (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))

Theoremnfbid 1496 If in a context 𝑥 is not free in 𝜓 and 𝜒, it is not free in (𝜓𝜒). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))

Theoremnfbi 1497 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)

1.3.7  The existential quantifier

Theorem19.8a 1498 If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
(𝜑 → ∃𝑥𝜑)

Theorem19.23bi 1499 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(∃𝑥𝜑𝜓)       (𝜑𝜓)

Theoremexlimih 1500 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(𝜓 → ∀𝑥𝜓)    &   (𝜑𝜓)       (∃𝑥𝜑𝜓)

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