P. 170 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 16901-17000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-fadc 16901	A refutable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
|- ( -. ph -> DECID ph )
Theorem	bj-dcfal 16902	The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
|- DECID F.
Theorem	bj-dcstab 16903	A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
|- (DECID ph -> STAB ph )
Theorem	bj-nnbidc 16904	If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 16891. (Contributed by BJ, 24-Nov-2023.)
|- ( -. -. ph -> (DECID ph <-> ph ) )
Theorem	bj-nndcALT 16905	Alternate proof of nndc 863. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by BJ, 9-Oct-2019.)
|- -. -. DECID ph
Theorem	bj-dcdc 16906	Decidability of a proposition is decidable if and only if that proposition is decidable. DECID is idempotent. (Contributed by BJ, 9-Oct-2019.)
|- (DECID DECID ph <-> DECID ph )
Theorem	bj-stdc 16907	Decidability of a proposition is stable if and only if that proposition is decidable. In particular, the assumption that every formula is stable implies that every formula is decidable, hence classical logic. (Contributed by BJ, 9-Oct-2019.)
|- (STAB DECID ph <-> DECID ph )
Theorem	bj-dcst 16908	Stability of a proposition is decidable if and only if that proposition is stable. (Contributed by BJ, 24-Nov-2023.)
|- (DECID STAB ph <-> STAB ph )
14.3.2  Predicate calculus
Theorem	bj-ex 16909*	Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1651 and 19.9ht 1694 or 19.23ht 1550). (Proof modification is discouraged.)
|- ( E. x ph -> ph )
Theorem	bj-hbalt 16910	Closed form of hbal 1530 (copied from set.mm). (Contributed by BJ, 2-May-2019.)
|- ( A. y ( ph -> A. x ph ) -> ( A. y ph -> A. x A. y ph ) )
Theorem	bj-nfalt 16911	Closed form of nfal 1629 (copied from set.mm). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)
|- ( A. x F/ y ph -> F/ y A. x ph )
Theorem	spimd 16912	Deduction form of spim 1791. (Contributed by BJ, 17-Oct-2019.)
|- ( ph -> F/ x ch )   &    |- ( ph -> A. x ( x = y -> ( ps -> ch ) ) )   =>    |- ( ph -> ( A. x ps -> ch ) )
Theorem	2spim 16913*	Double substitution, as in spim 1791. (Contributed by BJ, 17-Oct-2019.)
|- F/ x ch   &    |- F/ z ch   &    |- ( ( x = y /\ z = t ) -> ( ps -> ch ) )   =>    |- ( A. z A. x ps -> ch )
Theorem	ch2var 16914*	Implicit substitution of y for x and t for z into a theorem. (Contributed by BJ, 17-Oct-2019.)
|- F/ x ps   &    |- F/ z ps   &    |- ( ( x = y /\ z = t ) -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	ch2varv 16915*	Version of ch2var 16914 with nonfreeness hypotheses replaced with disjoint variable conditions. (Contributed by BJ, 17-Oct-2019.)
|- ( ( x = y /\ z = t ) -> ( ph <-> ps ) )   &    |- ph   =>    |- ps
Theorem	bj-exlimmp 16916	Lemma for bj-vtoclgf 16923. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- F/ x ps   &    |- ( ch -> ph )   =>    |- ( A. x ( ch -> ( ph -> ps ) ) -> ( E. x ch -> ps ) )
Theorem	bj-exlimmpi 16917	Lemma for bj-vtoclgf 16923. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
|- F/ x ps   &    |- ( ch -> ph )   &    |- ( ch -> ( ph -> ps ) )   =>    |- ( E. x ch -> ps )
Theorem	bj-sbimedh 16918	A strengthening of sbiedh 1840 (same proof). (Contributed by BJ, 16-Dec-2019.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ch -> A. x ch ) )   &    |- ( ph -> ( x = y -> ( ps -> ch ) ) )   =>    |- ( ph -> ( [ y / x ] ps -> ch ) )
Theorem	bj-sbimeh 16919	A strengthening of sbieh 1843 (same proof). (Contributed by BJ, 16-Dec-2019.)
|- ( ps -> A. x ps )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( [ y / x ] ph -> ps )
Theorem	bj-sbime 16920	A strengthening of sbie 1844 (same proof). (Contributed by BJ, 16-Dec-2019.)
|- F/ x ps   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( [ y / x ] ph -> ps )
14.3.3  Set theorey miscellaneous
Theorem	bj-el2oss1o 16921	Shorter proof of el2oss1o 6716 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( A e. 2o -> A C_ 1o )
14.3.4  Extensionality Various utility theorems using FOL and extensionality.
Theorem	bj-vtoclgft 16922	Weakening two hypotheses of vtoclgf 2881. (Contributed by BJ, 21-Nov-2019.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ph )   =>    |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. V -> ps ) )
Theorem	bj-vtoclgf 16923	Weakening two hypotheses of vtoclgf 2881. (Contributed by BJ, 21-Nov-2019.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ph )   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. V -> ps )
Theorem	elabgf0 16924	Lemma for elabgf 2968. (Contributed by BJ, 21-Nov-2019.)
|- ( x = A -> ( A e. { x | ph } <-> ph ) )
Theorem	elabgft1 16925	One implication of elabgf 2968, in closed form. (Contributed by BJ, 21-Nov-2019.)
|- F/_ x A   &    |- F/ x ps   =>    |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. { x | ph } -> ps ) )
Theorem	elabgf1 16926	One implication of elabgf 2968. (Contributed by BJ, 21-Nov-2019.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. { x | ph } -> ps )
Theorem	elabgf2 16927	One implication of elabgf 2968. (Contributed by BJ, 21-Nov-2019.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( A e. B -> ( ps -> A e. { x | ph } ) )
Theorem	elabf1 16928*	One implication of elabf 2969. (Contributed by BJ, 21-Nov-2019.)
|- F/ x ps   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. { x | ph } -> ps )
Theorem	elabf2 16929*	One implication of elabf 2969. (Contributed by BJ, 21-Nov-2019.)
|- F/ x ps   &    |- A e. _V   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( ps -> A e. { x | ph } )
Theorem	elab1 16930*	One implication of elab 2970. (Contributed by BJ, 21-Nov-2019.)
|- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. { x | ph } -> ps )
Theorem	elab2a 16931*	One implication of elab 2970. (Contributed by BJ, 21-Nov-2019.)
|- A e. _V   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( ps -> A e. { x | ph } )
Theorem	elabg2 16932*	One implication of elabg 2972. (Contributed by BJ, 21-Nov-2019.)
|- ( x = A -> ( ps -> ph ) )   =>    |- ( A e. V -> ( ps -> A e. { x | ph } ) )
Theorem	bj-rspgt 16933	Restricted specialization, generalized. Weakens a hypothesis of rspccv 2926 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
|- F/_ x A   &    |- F/_ x B   &    |- F/ x ps   =>    |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A. x e. B ph -> ( A e. B -> ps ) ) )
Theorem	bj-rspg 16934	Restricted specialization, generalized. Weakens a hypothesis of rspccv 2926 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
|- F/_ x A   &    |- F/_ x B   &    |- F/ x ps   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A. x e. B ph -> ( A e. B -> ps ) )
Theorem	cbvrald 16935*	Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.)
|- F/ x ph   &    |- F/ y ph   &    |- ( ph -> F/ y ps )   &    |- ( ph -> F/ x ch )   &    |- ( ph -> ( x = y -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) )
Theorem	bj-intabssel 16936	Version of intss1 3985 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
|- F/_ x A   =>    |- ( A e. V -> ( [. A / x ]. ph -> |^| { x | ph } C_ A ) )
Theorem	bj-intabssel1 16937	Version of intss1 3985 using a class abstraction and implicit substitution. Closed form of intmin3 3997. (Contributed by BJ, 29-Nov-2019.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( A e. V -> ( ps -> |^| { x | ph } C_ A ) )
Theorem	bj-elssuniab 16938	Version of elssuni 3963 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
|- F/_ x A   =>    |- ( A e. V -> ( [. A / x ]. ph -> A C_ U. { x | ph } ) )
Theorem	bj-sseq 16939	If two converse inclusions are characterized each by a formula, then equality is characterized by the conjunction of these formulas. (Contributed by BJ, 30-Nov-2019.)
|- ( ph -> ( ps <-> A C_ B ) )   &    |- ( ph -> ( ch <-> B C_ A ) )   =>    |- ( ph -> ( ( ps /\ ch ) <-> A = B ) )
14.3.5  Decidability of classes The question of decidability is essential in intuitionistic logic. In intuitionistic set theories, it is natural to define decidability of a set (or class) as decidability of membership in it. One can parameterize this notion with another set (or class) since it is often important to assess decidability of membership in one class among elements of another class. Namely, one will say that " A is decidable in B " if A. x e. BDECID x e. A (see df-dcin 16941). Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 16986).
Syntax	wdcin 16940	Syntax for decidability of a class in another.
wff A DECIDin B
Definition	df-dcin 16941*	Define decidability of a class in another. (Contributed by BJ, 19-Feb-2022.)
|- ( A DECIDin B <-> A. x e. B DECID x e. A )
Theorem	decidi 16942	Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
|- ( A DECIDin B -> ( X e. B -> ( X e. A \/ -. X e. A ) ) )
Theorem	decidr 16943*	Sufficient condition for being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
|- ( ph -> ( x e. B -> ( x e. A \/ -. x e. A ) ) )   =>    |- ( ph -> A DECIDin B )
Theorem	decidin 16944	If A is a decidable subclass of B (meaning: it is a subclass of B and it is decidable in B), and B is decidable in C, then A is decidable in C. (Contributed by BJ, 19-Feb-2022.)
|- ( ph -> A C_ B )   &    |- ( ph -> A DECIDin B )   &    |- ( ph -> B DECIDin C )   =>    |- ( ph -> A DECIDin C )
Theorem	uzdcinzz 16945	An upperset of integers is decidable in the integers. Reformulation of eluzdc 10019. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.)
|- ( M e. ZZ -> ( ZZ>= ` M ) DECIDin ZZ )
Theorem	sumdc2 16946*	Alternate proof of sumdc 12140, without disjoint variable condition on N , x (longer because the statement is taylored to the proof sumdc 12140). (Contributed by BJ, 19-Feb-2022.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> A C_ ( ZZ>= ` M ) )   &    |- ( ph -> A. x e. ( ZZ>= ` M )DECID x e. A )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> DECID N e. A )
14.3.6  Disjoint union
Theorem	djucllem 16947*	Lemma for djulcl 7391 and djurcl 7392. (Contributed by BJ, 4-Jul-2022.)
|- X e. _V   &    |- F = ( x e. _V |-> <. X , x >. )   =>    |- ( A e. B -> ( ( F |` B ) ` A ) e. ( { X } X. B ) )
Theorem	djulclALT 16948	Shortening of djulcl 7391 using djucllem 16947. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( C e. A -> ( (inl |` A ) ` C ) e. ( A ⊔ B ) )
Theorem	djurclALT 16949	Shortening of djurcl 7392 using djucllem 16947. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( C e. B -> ( (inr |` B ) ` C ) e. ( A ⊔ B ) )
14.3.7  Miscellaneous
Theorem	funmptd 16950	The maps-to notation defines a function (deduction form). Note: one should similarly prove a deduction form of funopab4 5414, then prove funmptd 16950 from it, and then prove funmpt 5415 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.)
|- ( ph -> F = ( x e. A |-> B ) )   =>    |- ( ph -> Fun F )
Theorem	fnmptd 16951*	The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.)
|- ( ph -> F = ( x e. A |-> B ) )   &    |- ( ( ph /\ x e. A ) -> B e. V )   =>    |- ( ph -> F Fn A )
Theorem	bj-charfun 16952*	Properties of the characteristic function on the class X of the class A. (Contributed by BJ, 15-Aug-2024.)
|- ( ph -> F = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )   =>    |- ( ph -> ( ( F : X --> ~P 1o /\ ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) : ( ( X i^i A ) u. ( X \ A ) ) --> 2o ) /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) )
Theorem	bj-charfundc 16953*	Properties of the characteristic function on the class X of the class A, provided membership in A is decidable in X. (Contributed by BJ, 6-Aug-2024.)
|- ( ph -> F = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )   &    |- ( ph -> A. x e. X DECID x e. A )   =>    |- ( ph -> ( F : X --> 2o /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) )
Theorem	bj-charfundcALT 16954*	Alternate proof of bj-charfundc 16953. It was expected to be much shorter since it uses bj-charfun 16952 for the main part of the proof and the rest is basic computations, but these turn out to be lengthy, maybe because of the limited library of available lemmas. (Contributed by BJ, 15-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ph -> F = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )   &    |- ( ph -> A. x e. X DECID x e. A )   =>    |- ( ph -> ( F : X --> 2o /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) )
Theorem	bj-charfunr 16955*	If a class A has a "weak" characteristic function on a class X, then negated membership in A is decidable (in other words, membership in A is testable) in X. The hypothesis imposes that X be a set. As usual, it could be formulated as |- ( ph -> ( F : X --> om /\ ... ) ) to deal with general classes, but that extra generality would not make the theorem much more useful. The theorem would still hold if the codomain of f were any class with testable equality to the point where ( X \ A ) is sent. (Contributed by BJ, 6-Aug-2024.)
|- ( ph -> E. f e. ( om ^m X ) ( A. x e. ( X i^i A ) ( f ` x ) =/= (/) /\ A. x e. ( X \ A ) ( f ` x ) = (/) ) )   =>    |- ( ph -> A. x e. X DECID -. x e. A )
Theorem	bj-charfunbi 16956*	In an ambient set X, if membership in A is stable, then it is decidable if and only if A has a characteristic function. This characterization can be applied to singletons when the set X has stable equality, which is the case as soon as it has a tight apartness relation. (Contributed by BJ, 6-Aug-2024.)
|- ( ph -> X e. V )   &    |- ( ph -> A. x e. X STAB x e. A )   =>    |- ( ph -> ( A. x e. X DECID x e. A <-> E. f e. ( 2o ^m X ) ( A. x e. ( X i^i A ) ( f ` x ) = 1o /\ A. x e. ( X \ A ) ( f ` x ) = (/) ) ) )
14.3.8  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes This section develops constructive Zermelo--Fraenkel set theory (CZF) on top of intuitionistic logic. It is a constructive theory in the sense that its logic is intuitionistic and it is predicative. "Predicative" means that new sets can be constructed only from already constructed sets. In particular, the axiom of separation ax-sep 4249 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 17029. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 4246 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 17127 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 17086. Similarly, the axiom of powerset ax-pow 4311 is not predicative (checking whether a set is included in another requires to universally quantifier over that "not yet constructed" set) and is replaced in CZF by the axiom of fullness or the axiom of subset collection ax-sscoll 17132. In an intuitionistic context, the axiom of regularity is stated in IZF as well as in CZF as the axiom of set induction ax-setind 4684. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 17113. For more details on CZF, a useful set of notes is Peter Aczel and Michael Rathjen, CST Book draft. (available at http://www1.maths.leeds.ac.uk/~rathjen/book.pdf 17113) and an interesting article is Michael Shulman, Comparing material and structural set theories, Annals of Pure and Applied Logic, Volume 170, Issue 4 (Apr. 2019), 465--504. https://doi.org/10.48550/arXiv.1808.05204 17113 I also thank Michael Rathjen and Michael Shulman for useful hints in the formulation of some results.
14.3.8.1  Bounded formulas The present definition of bounded formulas emerged from a discussion on GitHub between Jim Kingdon, Mario Carneiro and I, started 23-Sept-2019 (see https://github.com/metamath/set.mm/issues/1173 and links therein). In order to state certain axiom schemes of Constructive Zermelo–Fraenkel (CZF) set theory, like the axiom scheme of bounded (or restricted, or Δ0) separation, it is necessary to distinguish certain formulas, called bounded (or restricted, or Δ0) formulas. The necessity of considering bounded formulas also arises in several theories of bounded arithmetic, both classical or intuitionistic, for instance to state the axiom scheme of Δ0-induction. To formalize this in Metamath, there are several choices to make. A first choice is to either create a new type for bounded formulas, or to create a predicate on formulas that indicates whether they are bounded. In the first case, one creates a new type "wff0" with a new set of metavariables (ph0 ...) and an axiom "$a wff ph0 " ensuring that bounded formulas are formulas, so that one can reuse existing theorems, and then axioms take the form "$a wff0 ( ph0 -> ps0 )", etc. In the second case, one introduces a predicate "BOUNDED " with the intended meaning that "BOUNDED ph " is a formula meaning that ph is a bounded formula. We choose the second option, since the first would complicate the grammar, risking to make it ambiguous. (TODO: elaborate.) A second choice is to view "bounded" either as a syntactic or a semantic property. For instance, A. x T. is not syntactically bounded since it has an unbounded universal quantifier, but it is semantically bounded since it is equivalent to T. which is bounded. We choose the second option, so that formulas using defined symbols can be proved bounded. A third choice is in the form of the axioms, either in closed form or in inference form. One cannot state all the axioms in closed form, especially ax-bd0 16958. Indeed, if we posited it in closed form, then we could prove for instance |- ( ph -> BOUNDED ph ) and |- ( -. ph -> BOUNDED ph ) which is problematic (with the law of excluded middle, this would entail that all formulas are bounded, but even without it, too many formulas could be proved bounded...). (TODO: elaborate.) Having ax-bd0 16958 in inference form ensures that a formula can be proved bounded only if it is equivalent *for all values of the free variables* to a syntactically bounded one. The other axioms (ax-bdim 16959 through ax-bdsb 16967) can be written either in closed or inference form. The fact that ax-bd0 16958 is an inference is enough to ensure that the closed forms cannot be "exploited" to prove that some unbounded formulas are bounded. (TODO: check.) However, we state all the axioms in inference form to make it clear that we do not exploit any over-permissiveness. Finally, note that our logic has no terms, only variables. Therefore, we cannot prove for instance that x e. om is a bounded formula. However, since om can be defined as "the y such that PHI" a proof using the fact that x e. om is bounded can be converted to a proof in iset.mm by replacing om with y everywhere and prepending the antecedent PHI, since x e. y is bounded by ax-bdel 16966. For a similar method, see bj-omtrans 17101. Note that one cannot add an axiom |- BOUNDED x e. A since by bdph 16995 it would imply that every formula is bounded.
Syntax	wbd 16957	Syntax for the predicate BOUNDED.
wff BOUNDED ph
Axiom	ax-bd0 16958	If two formulas are equivalent, then boundedness of one implies boundedness of the other. (Contributed by BJ, 3-Oct-2019.)
|- ( ph <-> ps )   =>    |- (BOUNDED ph -> BOUNDED ps )
Axiom	ax-bdim 16959	An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph -> ps )
Axiom	ax-bdan 16960	The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph /\ ps )
Axiom	ax-bdor 16961	The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph \/ ps )
Axiom	ax-bdn 16962	The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   =>    |- BOUNDED -. ph
Axiom	ax-bdal 16963*	A bounded universal quantification of a bounded formula is bounded. Note the disjoint variable condition on x , y. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   =>    |- BOUNDED A. x e. y ph
Axiom	ax-bdex 16964*	A bounded existential quantification of a bounded formula is bounded. Note the disjoint variable condition on x , y. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   =>    |- BOUNDED E. x e. y ph
Axiom	ax-bdeq 16965	An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x = y
Axiom	ax-bdel 16966	An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x e. y
Axiom	ax-bdsb 16967	A formula resulting from proper substitution in a bounded formula is bounded. This probably cannot be proved from the other axioms, since neither the definiens in df-sb 1816, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED [ y / x ] ph
Theorem	bdeq 16968	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
|- ( ph <-> ps )   =>    |- (BOUNDED ph <-> BOUNDED ps )
Theorem	bd0 16969	A formula equivalent to a bounded one is bounded. See also bd0r 16970. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- ( ph <-> ps )   =>    |- BOUNDED ps
Theorem	bd0r 16970	A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 16969) biconditional in the hypothesis, to work better with definitions ( ps is the definiendum that one wants to prove bounded). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- ( ps <-> ph )   =>    |- BOUNDED ps
Theorem	bdbi 16971	A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph <-> ps )
Theorem	bdstab 16972	Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED STAB ph
Theorem	bddc 16973	Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED DECID ph
Theorem	bd3or 16974	A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   &    |- BOUNDED ch   =>    |- BOUNDED ( ph \/ ps \/ ch )
Theorem	bd3an 16975	A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   &    |- BOUNDED ch   =>    |- BOUNDED ( ph /\ ps /\ ch )
Theorem	bdth 16976	A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
|- ph   =>    |- BOUNDED ph
Theorem	bdtru 16977	The truth value T. is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED T.
Theorem	bdfal 16978	The truth value F. is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED F.
Theorem	bdnth 16979	A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
|- -. ph   =>    |- BOUNDED ph
Theorem	bdnthALT 16980	Alternate proof of bdnth 16979 not using bdfal 16978. Then, bdfal 16978 can be proved from this theorem, using fal 1409. The total number of proof steps would be 17 (for bdnthALT 16980) + 3 = 20, which is more than 8 (for bdfal 16978) + 9 (for bdnth 16979) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- -. ph   =>    |- BOUNDED ph
Theorem	bdxor 16981	The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph \/_ ps )
Theorem	bj-bdcel 16982*	Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)
|- BOUNDED y = A   =>    |- BOUNDED A e. x
Theorem	bdab 16983	Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED x e. { y | ph }
Theorem	bdcdeq 16984	Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED CondEq ( x = y -> ph )
14.3.8.2  Bounded classes In line with our definitions of classes as extensions of predicates, it is useful to define a predicate for bounded classes, which is done in df-bdc 16986. Note that this notion is only a technical device which can be used to shorten proofs of (semantic) boundedness of formulas. As will be clear by the end of this subsection (see for instance bdop 17020), one can prove the boundedness of any concrete term using only setvars and bounded formulas, for instance, |- BOUNDED ph => |- BOUNDED <. { x | ph } , ( { y , suc z } X. <. t , (/) >. ) >.. The proofs are long since one has to prove boundedness at each step of the construction, without being able to prove general theorems like |- BOUNDED A => |- BOUNDED { A }.
Syntax	wbdc 16985	Syntax for the predicate BOUNDED.
wff BOUNDED A
Definition	df-bdc 16986*	Define a bounded class as one such that membership in this class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
|- (BOUNDED A <-> A. xBOUNDED x e. A )
Theorem	bdceq 16987	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
|- A = B   =>    |- (BOUNDED A <-> BOUNDED B )
Theorem	bdceqi 16988	A class equal to a bounded one is bounded. Note the use of ax-ext 2220. See also bdceqir 16989. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- A = B   =>    |- BOUNDED B
Theorem	bdceqir 16989	A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 16988) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 16970). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- B = A   =>    |- BOUNDED B
Theorem	bdel 16990*	The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
|- (BOUNDED A -> BOUNDED x e. A )
Theorem	bdeli 16991*	Inference associated with bdel 16990. Its converse is bdelir 16992. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED x e. A
Theorem	bdelir 16992*	Inference associated with df-bdc 16986. Its converse is bdeli 16991. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x e. A   =>    |- BOUNDED A
Theorem	bdcv 16993	A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x
Theorem	bdcab 16994	A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED { x | ph }
Theorem	bdph 16995	A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.)
|- BOUNDED { x | ph }   =>    |- BOUNDED ph
Theorem	bds 16996*	Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 16967; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 16967. (Contributed by BJ, 19-Nov-2019.)
|- BOUNDED ph   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- BOUNDED ps
Theorem	bdcrab 16997*	A class defined by restricted abstraction from a bounded class and a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- BOUNDED ph   =>    |- BOUNDED { x e. A | ph }
Theorem	bdne 16998	Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED x =/= y
Theorem	bdnel 16999*	Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED x e/ A
Theorem	bdreu 17000*	Boundedness of existential uniqueness. Remark regarding restricted quantifiers: the formula A. x e. A ph need not be bounded even if A and ph are. Indeed, _V is bounded by bdcvv 17002, and |- ( A. x e. _V ph <-> A. x ph ) (in minimal propositional calculus), so by bd0 16969, if A. x e. _V ph were bounded when ph is bounded, then A. x ph would be bounded as well when ph is bounded, which is not the case. The same remark holds with E. , E! , E*. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED E! x e. y ph