P. 145 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14401-14500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ch2var 14401*	Implicit substitution of 𝑦 for 𝑥 and 𝑡 for 𝑧 into a theorem. (Contributed by BJ, 17-Oct-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑧𝜓    &   ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	ch2varv 14402*	Version of ch2var 14401 with nonfreeness hypotheses replaced with disjoint variable conditions. (Contributed by BJ, 17-Oct-2019.)
⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	bj-exlimmp 14403	Lemma for bj-vtoclgf 14410. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → 𝜑)    ⇒   ⊢ (∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜒 → 𝜓))
Theorem	bj-exlimmpi 14404	Lemma for bj-vtoclgf 14410. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → 𝜑)    &   ⊢ (𝜒 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)
Theorem	bj-sbimedh 14405	A strengthening of sbiedh 1787 (same proof). (Contributed by BJ, 16-Dec-2019.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → 𝜒))
Theorem	bj-sbimeh 14406	A strengthening of sbieh 1790 (same proof). (Contributed by BJ, 16-Dec-2019.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)
Theorem	bj-sbime 14407	A strengthening of sbie 1791 (same proof). (Contributed by BJ, 16-Dec-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)
12.2.3  Set theorey miscellaneous
Theorem	bj-el2oss1o 14408	Shorter proof of el2oss1o 6443 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 2o → 𝐴 ⊆ 1o)
12.2.4  Extensionality Various utility theorems using FOL and extensionality.
Theorem	bj-vtoclgft 14409	Weakening two hypotheses of vtoclgf 2795. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → 𝜑)    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → 𝜓))
Theorem	bj-vtoclgf 14410	Weakening two hypotheses of vtoclgf 2795. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → 𝜑)    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜓)
Theorem	elabgf0 14411	Lemma for elabgf 2879. (Contributed by BJ, 21-Nov-2019.)
⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))
Theorem	elabgft1 14412	One implication of elabgf 2879, in closed form. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))
Theorem	elabgf1 14413	One implication of elabgf 2879. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)
Theorem	elabgf2 14414	One implication of elabgf 2879. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
Theorem	elabf1 14415*	One implication of elabf 2880. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)
Theorem	elabf2 14416*	One implication of elabf 2880. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})
Theorem	elab1 14417*	One implication of elab 2881. (Contributed by BJ, 21-Nov-2019.)
⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)
Theorem	elab2a 14418*	One implication of elab 2881. (Contributed by BJ, 21-Nov-2019.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})
Theorem	elabg2 14419*	One implication of elabg 2883. (Contributed by BJ, 21-Nov-2019.)
⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
Theorem	bj-rspgt 14420	Restricted specialization, generalized. Weakens a hypothesis of rspccv 2838 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓)))
Theorem	bj-rspg 14421	Restricted specialization, generalized. Weakens a hypothesis of rspccv 2838 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓))
Theorem	cbvrald 14422*	Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒))
Theorem	bj-intabssel 14423	Version of intss1 3859 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))
Theorem	bj-intabssel1 14424	Version of intss1 3859 using a class abstraction and implicit substitution. Closed form of intmin3 3871. (Contributed by BJ, 29-Nov-2019.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))
Theorem	bj-elssuniab 14425	Version of elssuni 3837 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → 𝐴 ⊆ ∪ {𝑥 ∣ 𝜑}))
Theorem	bj-sseq 14426	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.)
⊢ (𝜑 → (𝜓 ↔ 𝐴 ⊆ 𝐵))    &   ⊢ (𝜑 → (𝜒 ↔ 𝐵 ⊆ 𝐴))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ 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 ∀𝑥 ∈ 𝐵DECID 𝑥 ∈ 𝐴 (see df-dcin 14428). Note the similarity with the definition of a bounded class as a class for which membership in it is a bounded proposition (df-bdc 14475).
Syntax	wdcin 14427	Syntax for decidability of a class in another.
wff 𝐴 DECIDin 𝐵
Definition	df-dcin 14428*	Define decidability of a class in another. (Contributed by BJ, 19-Feb-2022.)
⊢ (𝐴 DECIDin 𝐵 ↔ ∀𝑥 ∈ 𝐵 DECID 𝑥 ∈ 𝐴)
Theorem	decidi 14429	Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
⊢ (𝐴 DECIDin 𝐵 → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐴 ∨ ¬ 𝑋 ∈ 𝐴)))
Theorem	decidr 14430*	Sufficient condition for being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
⊢ (𝜑 → (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐴)))    ⇒   ⊢ (𝜑 → 𝐴 DECIDin 𝐵)
Theorem	decidin 14431	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.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐴 DECIDin 𝐵)    &   ⊢ (𝜑 → 𝐵 DECIDin 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 DECIDin 𝐶)
Theorem	uzdcinzz 14432	An upperset of integers is decidable in the integers. Reformulation of eluzdc 9608. (Contributed by Jim Kingdon, 18-Apr-2020.) (Revised by BJ, 19-Feb-2022.)
⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) DECIDin ℤ)
Theorem	sumdc2 14433*	Alternate proof of sumdc 11361, without disjoint variable condition on 𝑁, 𝑥 (longer because the statement is taylored to the proof sumdc 11361). (Contributed by BJ, 19-Feb-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)DECID 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → DECID 𝑁 ∈ 𝐴)
12.2.6  Disjoint union
Theorem	djucllem 14434*	Lemma for djulcl 7049 and djurcl 7050. (Contributed by BJ, 4-Jul-2022.)
⊢ 𝑋 ∈ V    &   ⊢ 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)    ⇒   ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) ∈ ({𝑋} × 𝐵))
Theorem	djulclALT 14435	Shortening of djulcl 7049 using djucllem 14434. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴 ⊔ 𝐵))
Theorem	djurclALT 14436	Shortening of djurcl 7050 using djucllem 14434. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴 ⊔ 𝐵))
12.2.7  Miscellaneous
Theorem	funmptd 14437	The maps-to notation defines a function (deduction form). Note: one should similarly prove a deduction form of funopab4 5253, then prove funmptd 14437 from it, and then prove funmpt 5254 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    ⇒   ⊢ (𝜑 → Fun 𝐹)
Theorem	fnmptd 14438*	The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 Fn 𝐴)
Theorem	if0ab 14439*	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(𝜑, 𝐴, ∅) ⊆ 𝐴 and therefore, using elpwg 3583, ⊢ (𝐴 ∈ 𝑉 → if(𝜑, 𝐴, ∅) ∈ 𝒫 𝐴), from which fmelpw1o 14440 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)
⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑}
Theorem	fmelpw1o 14440	With a formula 𝜑 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 ⊤ and ⊥, by nndc 851, which translate to 1o and ∅ respectively by iftrue 3539 and iffalse 3542, giving pwtrufal 14629). As proved in if0ab 14439, the associated element of 𝒫 1o is the extension, in 𝒫 1o, of the formula 𝜑. (Contributed by BJ, 15-Aug-2024.)
⊢ if(𝜑, 1o, ∅) ∈ 𝒫 1o
Theorem	bj-charfun 14441*	Properties of the characteristic function on the class 𝑋 of the class 𝐴. (Contributed by BJ, 15-Aug-2024.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))    ⇒   ⊢ (𝜑 → ((𝐹:𝑋⟶𝒫 1o ∧ (𝐹 ↾ ((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))):((𝑋 ∩ 𝐴) ∪ (𝑋 ∖ 𝐴))⟶2o) ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))
Theorem	bj-charfundc 14442*	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.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))
Theorem	bj-charfundcALT 14443*	Alternate proof of bj-charfundc 14442. It was expected to be much shorter since it uses bj-charfun 14441 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.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅)))
Theorem	bj-charfunr 14444*	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 ⊢ (𝜑 → (𝐹:𝑋⟶ω ∧ ...)) 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.)
⊢ (𝜑 → ∃𝑓 ∈ (ω ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) ≠ ∅ ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID ¬ 𝑥 ∈ 𝐴)
Theorem	bj-charfunbi 14445*	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.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 STAB 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴 ↔ ∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (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 4121 is not predicative (because we cannot allow all formulas to define a subset) and is replaced in CZF by bounded separation ax-bdsep 14518. Because this axiom is weaker than full separation, the axiom of replacement or collection ax-coll 4118 of ZF and IZF has to be strengthened in CZF to the axiom of strong collection ax-strcoll 14616 (which is a theorem of IZF), and the axiom of infinity needs a more precise version, the von Neumann axiom of infinity ax-infvn 14575. Similarly, the axiom of powerset ax-pow 4174 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 14621. 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 4536. It is sometimes interesting to study the weakening of CZF where that axiom is replaced by bounded set induction ax-bdsetind 14602. 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 14602) 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 14602 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 𝜑 " is a formula meaning that 𝜑 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, ∀𝑥⊤ is not syntactically bounded since it has an unbounded universal quantifier, but it is semantically bounded since it is equivalent to ⊤ 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 14447. Indeed, if we posited it in closed form, then we could prove for instance ⊢ (𝜑 → BOUNDED 𝜑) and ⊢ (¬ 𝜑 → BOUNDED 𝜑) 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 14447 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 14448 through ax-bdsb 14456) can be written either in closed or inference form. The fact that ax-bd0 14447 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 ∈ ω is a bounded formula. However, since ω can be defined as "the 𝑦 such that PHI" a proof using the fact that 𝑥 ∈ ω is bounded can be converted to a proof in iset.mm by replacing ω with 𝑦 everywhere and prepending the antecedent PHI, since 𝑥 ∈ 𝑦 is bounded by ax-bdel 14455. For a similar method, see bj-omtrans 14590. Note that one cannot add an axiom ⊢ BOUNDED 𝑥 ∈ 𝐴 since by bdph 14484 it would imply that every formula is bounded.
Syntax	wbd 14446	Syntax for the predicate BOUNDED.
wff BOUNDED 𝜑
Axiom	ax-bd0 14447	If two formulas are equivalent, then boundedness of one implies boundedness of the other. (Contributed by BJ, 3-Oct-2019.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (BOUNDED 𝜑 → BOUNDED 𝜓)
Axiom	ax-bdim 14448	An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 → 𝜓)
Axiom	ax-bdan 14449	The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ∧ 𝜓)
Axiom	ax-bdor 14450	The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ∨ 𝜓)
Axiom	ax-bdn 14451	The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ¬ 𝜑
Axiom	ax-bdal 14452*	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 𝜑    ⇒   ⊢ BOUNDED ∀𝑥 ∈ 𝑦 𝜑
Axiom	ax-bdex 14453*	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 𝜑    ⇒   ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑
Axiom	ax-bdeq 14454	An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥 = 𝑦
Axiom	ax-bdel 14455	An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥 ∈ 𝑦
Axiom	ax-bdsb 14456	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 𝜑    ⇒   ⊢ BOUNDED [𝑦 / 𝑥]𝜑
Theorem	bdeq 14457	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (BOUNDED 𝜑 ↔ BOUNDED 𝜓)
Theorem	bd0 14458	A formula equivalent to a bounded one is bounded. See also bd0r 14459. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ BOUNDED 𝜓
Theorem	bd0r 14459	A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 14458) biconditional in the hypothesis, to work better with definitions (𝜓 is the definiendum that one wants to prove bounded). (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ (𝜓 ↔ 𝜑)    ⇒   ⊢ BOUNDED 𝜓
Theorem	bdbi 14460	A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ↔ 𝜓)
Theorem	bdstab 14461	Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED STAB 𝜑
Theorem	bddc 14462	Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED DECID 𝜑
Theorem	bd3or 14463	A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    &   ⊢ BOUNDED 𝜒    ⇒   ⊢ BOUNDED (𝜑 ∨ 𝜓 ∨ 𝜒)
Theorem	bd3an 14464	A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    &   ⊢ BOUNDED 𝜒    ⇒   ⊢ BOUNDED (𝜑 ∧ 𝜓 ∧ 𝜒)
Theorem	bdth 14465	A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
⊢ 𝜑    ⇒   ⊢ BOUNDED 𝜑
Theorem	bdtru 14466	The truth value ⊤ is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED ⊤
Theorem	bdfal 14467	The truth value ⊥ is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED ⊥
Theorem	bdnth 14468	A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
⊢ ¬ 𝜑    ⇒   ⊢ BOUNDED 𝜑
Theorem	bdnthALT 14469	Alternate proof of bdnth 14468 not using bdfal 14467. Then, bdfal 14467 can be proved from this theorem, using fal 1360. The total number of proof steps would be 17 (for bdnthALT 14469) + 3 = 20, which is more than 8 (for bdfal 14467) + 9 (for bdnth 14468) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ 𝜑    ⇒   ⊢ BOUNDED 𝜑
Theorem	bdxor 14470	The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    &   ⊢ BOUNDED 𝜓    ⇒   ⊢ BOUNDED (𝜑 ⊻ 𝜓)
Theorem	bj-bdcel 14471*	Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)
⊢ BOUNDED 𝑦 = 𝐴    ⇒   ⊢ BOUNDED 𝐴 ∈ 𝑥
Theorem	bdab 14472	Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED 𝑥 ∈ {𝑦 ∣ 𝜑}
Theorem	bdcdeq 14473	Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED CondEq(𝑥 = 𝑦 → 𝜑)
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 14475. 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 14509), one can prove the boundedness of any concrete term using only setvars and bounded formulas, for instance, ⊢ BOUNDED 𝜑 ⇒ ⊢ BOUNDED ⟨{𝑥 ∣ 𝜑}, ({𝑦, suc 𝑧} × ⟨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 14474	Syntax for the predicate BOUNDED.
wff BOUNDED 𝐴
Definition	df-bdc 14475*	Define a bounded class as one such that membership in this class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
⊢ (BOUNDED 𝐴 ↔ ∀𝑥BOUNDED 𝑥 ∈ 𝐴)
Theorem	bdceq 14476	Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (BOUNDED 𝐴 ↔ BOUNDED 𝐵)
Theorem	bdceqi 14477	A class equal to a bounded one is bounded. Note the use of ax-ext 2159. See also bdceqir 14478. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐴 = 𝐵    ⇒   ⊢ BOUNDED 𝐵
Theorem	bdceqir 14478	A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 14477) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 14459). (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ 𝐵 = 𝐴    ⇒   ⊢ BOUNDED 𝐵
Theorem	bdel 14479*	The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
⊢ (BOUNDED 𝐴 → BOUNDED 𝑥 ∈ 𝐴)
Theorem	bdeli 14480*	Inference associated with bdel 14479. Its converse is bdelir 14481. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED 𝑥 ∈ 𝐴
Theorem	bdelir 14481*	Inference associated with df-bdc 14475. Its converse is bdeli 14480. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥 ∈ 𝐴    ⇒   ⊢ BOUNDED 𝐴
Theorem	bdcv 14482	A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝑥
Theorem	bdcab 14483	A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED {𝑥 ∣ 𝜑}
Theorem	bdph 14484	A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.)
⊢ BOUNDED {𝑥 ∣ 𝜑}    ⇒   ⊢ BOUNDED 𝜑
Theorem	bds 14485*	Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 14456; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 14456. (Contributed by BJ, 19-Nov-2019.)
⊢ BOUNDED 𝜑    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ BOUNDED 𝜓
Theorem	bdcrab 14486*	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 𝜑    ⇒   ⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑}
Theorem	bdne 14487	Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝑥 ≠ 𝑦
Theorem	bdnel 14488*	Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED 𝑥 ∉ 𝐴
Theorem	bdreu 14489*	Boundedness of existential uniqueness. Remark regarding restricted quantifiers: the formula ∀𝑥 ∈ 𝐴𝜑 need not be bounded even if 𝐴 and 𝜑 are. Indeed, V is bounded by bdcvv 14491, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 14458, if ∀𝑥 ∈ V𝜑 were bounded when 𝜑 is bounded, then ∀𝑥𝜑 would be bounded as well when 𝜑 is bounded, which is not the case. The same remark holds with ∃, ∃!, ∃*. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑
Theorem	bdrmo 14490*	Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑
Theorem	bdcvv 14491	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 14492	A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 14493. (Contributed by BJ, 16-Oct-2019.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED [𝑦 / 𝑥]𝜑
Theorem	bdsbcALT 14493	Alternate proof of bdsbc 14492. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ BOUNDED 𝜑    ⇒   ⊢ BOUNDED [𝑦 / 𝑥]𝜑
Theorem	bdccsb 14494	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 14495	The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ BOUNDED (𝐴 ∖ 𝐵)
Theorem	bdcun 14496	The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ BOUNDED (𝐴 ∪ 𝐵)
Theorem	bdcin 14497	The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    &   ⊢ BOUNDED 𝐵    ⇒   ⊢ BOUNDED (𝐴 ∩ 𝐵)
Theorem	bdss 14498	The inclusion of a setvar in a bounded class is a bounded formula. Note: apparently, we cannot prove from the present axioms that equality of two bounded classes is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED 𝐴    ⇒   ⊢ BOUNDED 𝑥 ⊆ 𝐴
Theorem	bdcnul 14499	The empty class is bounded. See also bdcnulALT 14500. (Contributed by BJ, 3-Oct-2019.)
⊢ BOUNDED ∅
Theorem	bdcnulALT 14500	Alternate proof of bdcnul 14499. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 14478, or use the corresponding characterizations of its elements followed by bdelir 14481. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ BOUNDED ∅