P. 147 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14601-14700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-nfalt 14601	Closed form of nfal 1576 (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 14602	Deduction form of spim 1738. (Contributed by BJ, 17-Oct-2019.)
|- ( ph -> F/ x ch )   &    |- ( ph -> A. x ( x = y -> ( ps -> ch ) ) )   =>    |- ( ph -> ( A. x ps -> ch ) )
Theorem	2spim 14603*	Double substitution, as in spim 1738. (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 14604*	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 14605*	Version of ch2var 14604 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 14606	Lemma for bj-vtoclgf 14613. (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 14607	Lemma for bj-vtoclgf 14613. (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 14608	A strengthening of sbiedh 1787 (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 14609	A strengthening of sbieh 1790 (same proof). (Contributed by BJ, 16-Dec-2019.)
|- ( ps -> A. x ps )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( [ y / x ] ph -> ps )
Theorem	bj-sbime 14610	A strengthening of sbie 1791 (same proof). (Contributed by BJ, 16-Dec-2019.)
|- F/ x ps   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( [ y / x ] ph -> ps )
12.2.3  Set theorey miscellaneous
Theorem	bj-el2oss1o 14611	Shorter proof of el2oss1o 6446 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( A e. 2o -> A C_ 1o )
12.2.4  Extensionality Various utility theorems using FOL and extensionality.
Theorem	bj-vtoclgft 14612	Weakening two hypotheses of vtoclgf 2797. (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 14613	Weakening two hypotheses of vtoclgf 2797. (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 14614	Lemma for elabgf 2881. (Contributed by BJ, 21-Nov-2019.)
|- ( x = A -> ( A e. { x | ph } <-> ph ) )
Theorem	elabgft1 14615	One implication of elabgf 2881, 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 14616	One implication of elabgf 2881. (Contributed by BJ, 21-Nov-2019.)
|- F/_ x A   &    |- F/ x ps   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. { x | ph } -> ps )
Theorem	elabgf2 14617	One implication of elabgf 2881. (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 14618*	One implication of elabf 2882. (Contributed by BJ, 21-Nov-2019.)
|- F/ x ps   &    |- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. { x | ph } -> ps )
Theorem	elabf2 14619*	One implication of elabf 2882. (Contributed by BJ, 21-Nov-2019.)
|- F/ x ps   &    |- A e. _V   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( ps -> A e. { x | ph } )
Theorem	elab1 14620*	One implication of elab 2883. (Contributed by BJ, 21-Nov-2019.)
|- ( x = A -> ( ph -> ps ) )   =>    |- ( A e. { x | ph } -> ps )
Theorem	elab2a 14621*	One implication of elab 2883. (Contributed by BJ, 21-Nov-2019.)
|- A e. _V   &    |- ( x = A -> ( ps -> ph ) )   =>    |- ( ps -> A e. { x | ph } )
Theorem	elabg2 14622*	One implication of elabg 2885. (Contributed by BJ, 21-Nov-2019.)
|- ( x = A -> ( ps -> ph ) )   =>    |- ( A e. V -> ( ps -> A e. { x | ph } ) )
Theorem	bj-rspgt 14623	Restricted specialization, generalized. Weakens a hypothesis of rspccv 2840 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 14624	Restricted specialization, generalized. Weakens a hypothesis of rspccv 2840 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 14625*	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 14626	Version of intss1 3861 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 14627	Version of intss1 3861 using a class abstraction and implicit substitution. Closed form of intmin3 3873. (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 14628	Version of elssuni 3839 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 14629	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 ) )
12.2.5  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 14631). Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 14678).
Syntax	wdcin 14630	Syntax for decidability of a class in another.
wff A DECIDin B
Definition	df-dcin 14631*	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 14632	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 14633*	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 14634	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 14635	An upperset of integers is decidable in the integers. Reformulation of eluzdc 9612. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.)
|- ( M e. ZZ -> ( ZZ>= ` M ) DECIDin ZZ )
Theorem	sumdc2 14636*	Alternate proof of sumdc 11368, without disjoint variable condition on N , x (longer because the statement is taylored to the proof sumdc 11368). (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 )
12.2.6  Disjoint union
Theorem	djucllem 14637*	Lemma for djulcl 7052 and djurcl 7053. (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 14638	Shortening of djulcl 7052 using djucllem 14637. (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 14639	Shortening of djurcl 7053 using djucllem 14637. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( C e. B -> ( (inr |` B ) ` C ) e. ( A ⊔ B ) )
12.2.7  Miscellaneous
Theorem	funmptd 14640	The maps-to notation defines a function (deduction form). Note: one should similarly prove a deduction form of funopab4 5255, then prove funmptd 14640 from it, and then prove funmpt 5256 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 14641*	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	if0ab 14642*	Expression of a conditional class as a class abstraction when the False alternative is the empty class: in that case, the conditional class is the extension, in the True alternative, of the condition. Remark: a consequence which could be formalized is the inclusion |- if ( ph , A , (/) ) C_ A and therefore, using elpwg 3585, |- ( A e. V -> if ( ph , A , (/) ) e. ~P A ), from which fmelpw1o 14643 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)
|- if ( ph , A , (/) ) = { x e. A | ph }
Theorem	fmelpw1o 14643	With a formula ph one can associate an element of ~P 1o, which can therefore be thought of as the set of "truth values" (but recall that there are no other genuine truth values than T. and F., by nndc 851, which translate to 1o and (/) respectively by iftrue 3541 and iffalse 3544, giving pwtrufal 14832). As proved in if0ab 14642, the associated element of ~P 1o is the extension, in ~P 1o, of the formula ph. (Contributed by BJ, 15-Aug-2024.)
|- if ( ph , 1o , (/) ) e. ~P 1o
Theorem	bj-charfun 14644*	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 14645*	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 14646*	Alternate proof of bj-charfundc 14645. It was expected to be much shorter since it uses bj-charfun 14644 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 14647*	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 14648*	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 ) = (/) ) ) )
12.2.8  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 4123 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 14721. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 4120 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 14819 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 14778. Similarly, the axiom of powerset ax-pow 4176 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 14824. 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 4538. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 14805. 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 14805) 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 14805 I also thank Michael Rathjen and Michael Shulman for useful hints in the formulation of some results.
12.2.8.1  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 14650. 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 14650 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 14651 through ax-bdsb 14659) can be written either in closed or inference form. The fact that ax-bd0 14650 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 14658. For a similar method, see bj-omtrans 14793. Note that one cannot add an axiom |- BOUNDED x e. A since by bdph 14687 it would imply that every formula is bounded.
Syntax	wbd 14649	Syntax for the predicate BOUNDED.
wff BOUNDED ph
Axiom	ax-bd0 14650	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 14651	An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph -> ps )
Axiom	ax-bdan 14652	The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph /\ ps )
Axiom	ax-bdor 14653	The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph \/ ps )
Axiom	ax-bdn 14654	The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.)
|- BOUNDED ph   =>    |- BOUNDED -. ph
Axiom	ax-bdal 14655*	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 14656*	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 14657	An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x = y
Axiom	ax-bdel 14658	An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x e. y
Axiom	ax-bdsb 14659	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 1763, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED [ y / x ] ph
Theorem	bdeq 14660	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
|- ( ph <-> ps )   =>    |- (BOUNDED ph <-> BOUNDED ps )
Theorem	bd0 14661	A formula equivalent to a bounded one is bounded. See also bd0r 14662. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- ( ph <-> ps )   =>    |- BOUNDED ps
Theorem	bd0r 14662	A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 14661) 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 14663	A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   &    |- BOUNDED ps   =>    |- BOUNDED ( ph <-> ps )
Theorem	bdstab 14664	Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED STAB ph
Theorem	bddc 14665	Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED DECID ph
Theorem	bd3or 14666	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 14667	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 14668	A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
|- ph   =>    |- BOUNDED ph
Theorem	bdtru 14669	The truth value T. is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED T.
Theorem	bdfal 14670	The truth value F. is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED F.
Theorem	bdnth 14671	A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
|- -. ph   =>    |- BOUNDED ph
Theorem	bdnthALT 14672	Alternate proof of bdnth 14671 not using bdfal 14670. Then, bdfal 14670 can be proved from this theorem, using fal 1360. The total number of proof steps would be 17 (for bdnthALT 14672) + 3 = 20, which is more than 8 (for bdfal 14670) + 9 (for bdnth 14671) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- -. ph   =>    |- BOUNDED ph
Theorem	bdxor 14673	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 14674*	Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)
|- BOUNDED y = A   =>    |- BOUNDED A e. x
Theorem	bdab 14675	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 14676	Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED CondEq ( x = y -> ph )
12.2.8.2  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 14678. 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 14712), 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 14677	Syntax for the predicate BOUNDED.
wff BOUNDED A
Definition	df-bdc 14678*	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 14679	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
|- A = B   =>    |- (BOUNDED A <-> BOUNDED B )
Theorem	bdceqi 14680	A class equal to a bounded one is bounded. Note the use of ax-ext 2159. See also bdceqir 14681. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- A = B   =>    |- BOUNDED B
Theorem	bdceqir 14681	A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 14680) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 14662). (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- B = A   =>    |- BOUNDED B
Theorem	bdel 14682*	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 14683*	Inference associated with bdel 14682. Its converse is bdelir 14684. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED x e. A
Theorem	bdelir 14684*	Inference associated with df-bdc 14678. Its converse is bdeli 14683. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x e. A   =>    |- BOUNDED A
Theorem	bdcv 14685	A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED x
Theorem	bdcab 14686	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 14687	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 14688*	Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 14659; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 14659. (Contributed by BJ, 19-Nov-2019.)
|- BOUNDED ph   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- BOUNDED ps
Theorem	bdcrab 14689*	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 14690	Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED x =/= y
Theorem	bdnel 14691*	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 14692*	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 14694, and |- ( A. x e. _V ph <-> A. x ph ) (in minimal propositional calculus), so by bd0 14661, 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
Theorem	bdrmo 14693*	Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED E* x e. y ph
Theorem	bdcvv 14694	The universal class is bounded. The formulation may sound strange, but recall that here, "bounded" means "Δ0". (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED _V
Theorem	bdsbc 14695	A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 14696. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED ph   =>    |- BOUNDED [. y / x ]. ph
Theorem	bdsbcALT 14696	Alternate proof of bdsbc 14695. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
|- BOUNDED ph   =>    |- BOUNDED [. y / x ]. ph
Theorem	bdccsb 14697	A class resulting from proper substitution of a setvar for a setvar in a bounded class is bounded. (Contributed by BJ, 16-Oct-2019.)
|- BOUNDED A   =>    |- BOUNDED [_ y / x ]_ A
Theorem	bdcdif 14698	The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- BOUNDED B   =>    |- BOUNDED ( A \ B )
Theorem	bdcun 14699	The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- BOUNDED B   =>    |- BOUNDED ( A u. B )
Theorem	bdcin 14700	The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
|- BOUNDED A   &    |- BOUNDED B   =>    |- BOUNDED ( A i^i B )