P. 383 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38201-38300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ts3an2 38201	A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)))
Theorem	ts3an3 38202	A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (𝜒 ∨ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)))
Theorem	ts3or1 38203	A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (((𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ 𝜓 ∨ 𝜒)))
Theorem	ts3or2 38204	A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜓 ∨ 𝜒)))
Theorem	ts3or3 38205	A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
⊢ (𝜃 → (¬ 𝜒 ∨ (𝜑 ∨ 𝜓 ∨ 𝜒)))
21.25.3  Equality deductions A collection of theorems for commuting equalities (or biconditionals) with other constructs.
Theorem	iuneq2f 38206	Equality deduction for indexed union. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)
Theorem	rabeq12f 38207	Equality deduction for restricted class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜓})
Theorem	csbeq12 38208	Equality deduction for substitution in class. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐷)
Theorem	sbeqi 38209	Equality deduction for substitution. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ ((𝑥 = 𝑦 ∧ ∀𝑧(𝜑 ↔ 𝜓)) → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜓))
Theorem	ralbi12f 38210	Equality deduction for restricted universal quantification. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓))
Theorem	oprabbi 38211	Equality deduction for class abstraction of nested ordered pairs. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓})
Theorem	mpobi123f 38212*	Equality deduction for maps-to notations with two arguments. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑦𝐶    &   ⊢ Ⅎ𝑦𝐷    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ Ⅎ𝑥𝐷    ⇒   ⊢ (((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝐸 = 𝐹) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ 𝐹))
Theorem	iuneq12f 38213	Equality deduction for indexed unions. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)
Theorem	iineq12f 38214	Equality deduction for indexed intersections. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷)
Theorem	opabbi 38215	Equality deduction for class abstraction of ordered pairs. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
Theorem	mptbi12f 38216	Equality deduction for maps-to notations. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐷 = 𝐸) → (𝑥 ∈ 𝐴 ↦ 𝐷) = (𝑥 ∈ 𝐵 ↦ 𝐸))
21.25.4  Miscellanea Work in progress or things that do not belong anywhere else.
Theorem	orcomdd 38217	Commutativity of logic disjunction, in double deduction form. Should not be moved to main, see PR #3034 in Github. Use orcomd 871 instead. (Contributed by Giovanni Mascellani, 19-Mar-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 ∨ 𝜒)))
Theorem	scottexf 38218*	A version of scottex 9788 with nonfree variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
Theorem	scott0f 38219*	A version of scott0 9789 with nonfree variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 = ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
Theorem	scottn0f 38220*	A version of scott0f 38219 with inequalities instead of equalities. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ≠ ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅)
Theorem	ac6s3f 38221*	Generalization of the Axiom of Choice to classes, with bound-variable hypothesis. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
⊢ Ⅎ𝑦𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 → ∃𝑓∀𝑥 ∈ 𝐴 𝜓)
Theorem	ac6s6 38222*	Generalization of the Axiom of Choice to classes, moving the existence condition in the consequent. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
⊢ Ⅎ𝑦𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓)
Theorem	ac6s6f 38223*	Generalization of the Axiom of Choice to classes, moving the existence condition in the consequent. (Contributed by Giovanni Mascellani, 20-Aug-2018.)
⊢ 𝐴 ∈ V    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ ∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓)
21.26  Mathbox for Peter Mazsa
21.26.1  Notations
Syntax	cxrn 38224	Extend the definition of a class to include the range Cartesian product class.
class (𝐴 ⋉ 𝐵)
Syntax	cadjliftmap 38225	Extend the definition of a class to include the class of adjoined lift maps.
class (𝑅 AdjLiftMap 𝐴)
Syntax	cblockliftmap 38226	Extend the definition of a class to include the class of block lift maps.
class (𝑅 BlockLiftMap 𝐴)
Syntax	csucmap 38227	Extend the definition of a class to include the class of successor maps.
class SucMap
Syntax	csuccl 38228	Extend the definition of a class to include the class of successors.
class Suc
Syntax	cpre 38229	Extend the definition of a class to include the predecessor of a class.
class pre 𝑁
Syntax	cblockliftfix 38230	Extend the definition of a class to include the class of equilibrium block lifts.
class BlockLiftFix
Syntax	cshiftstable 38231	Extend the definition of a class to include the shift stability class.
class (𝑆 ShiftStable 𝐹)
Syntax	ccoss 38232	Extend the definition of a class to include the class of cosets by a class. (Read: the class of cosets by 𝑅.)
class ≀ 𝑅
Syntax	ccoels 38233	Extend the definition of a class to include the class of coelements on a class. (Read: the class of coelements on 𝐴.)
class ∼ 𝐴
Syntax	crels 38234	Extend the definition of a class to include the relation class.
class Rels
Syntax	cssr 38235	Extend the definition of a class to include the subset class.
class S
Syntax	crefs 38236	Extend the definition of a class to include the reflexivity class.
class Refs
Syntax	crefrels 38237	Extend the definition of a class to include the reflexive relations class.
class RefRels
Syntax	wrefrel 38238	Extend the definition of a wff to include the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.)
wff RefRel 𝑅
Syntax	ccnvrefs 38239	Extend the definition of a class to include the converse reflexivity class.
class CnvRefs
Syntax	ccnvrefrels 38240	Extend the definition of a class to include the converse reflexive relations class.
class CnvRefRels
Syntax	wcnvrefrel 38241	Extend the definition of a wff to include the converse reflexive relation predicate. (Read: 𝑅 is a converse reflexive relation.)
wff CnvRefRel 𝑅
Syntax	csyms 38242	Extend the definition of a class to include the symmetry class.
class Syms
Syntax	csymrels 38243	Extend the definition of a class to include the symmetry relations class.
class SymRels
Syntax	wsymrel 38244	Extend the definition of a wff to include the symmetry relation predicate. (Read: 𝑅 is a symmetric relation.)
wff SymRel 𝑅
Syntax	ctrs 38245	Extend the definition of a class to include the transitivity class (but cf. the transitive class defined in df-tr 5203).
class Trs
Syntax	ctrrels 38246	Extend the definition of a class to include the transitive relations class.
class TrRels
Syntax	wtrrel 38247	Extend the definition of a wff to include the transitive relation predicate. (Read: 𝑅 is a transitive relation.)
wff TrRel 𝑅
Syntax	ceqvrels 38248	Extend the definition of a class to include the equivalence relations class.
class EqvRels
Syntax	weqvrel 38249	Extend the definition of a wff to include the equivalence relation predicate. (Read: 𝑅 is an equivalence relation.)
wff EqvRel 𝑅
Syntax	ccoeleqvrels 38250	Extend the definition of a class to include the coelement equivalence relations class.
class CoElEqvRels
Syntax	wcoeleqvrel 38251	Extend the definition of a wff to include the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.)
wff CoElEqvRel 𝐴
Syntax	credunds 38252	Extend the definition of a class to include the redundancy class.
class Redunds
Syntax	wredund 38253	Extend the definition of a wff to include the redundancy predicate. (Read: 𝐴 is redundant with respect to 𝐵 in 𝐶.)
wff 𝐴 Redund ⟨𝐵, 𝐶⟩
Syntax	wredundp 38254	Extend wff definition to include the redundancy operator for propositions.
wff redund (𝜑, 𝜓, 𝜒)
Syntax	cdmqss 38255	Extend the definition of a class to include the domain quotients class.
class DomainQss
Syntax	wdmqs 38256	Extend the definition of a wff to include the domain quotient predicate. (Read: the domain quotient of 𝑅 is 𝐴.)
wff 𝑅 DomainQs 𝐴
Syntax	cers 38257	Extend the definition of a class to include the equivalence relations on their domain quotients class.
class Ers
Syntax	werALTV 38258	Extend the definition of a wff to include the equivalence relation on its domain quotient predicate. (Read: 𝑅 is an equivalence relation on its domain quotient 𝐴.)
wff 𝑅 ErALTV 𝐴
Syntax	ccomembers 38259	Extend the definition of a class to include the comember equivalence relations class.
class CoMembErs
Syntax	wcomember 38260	Extend the definition of a wff to include the comember equivalence relation predicate. (Read: the comember equivalence relation on 𝐴, or, the restricted coelement equivalence relation on its domain quotient 𝐴.)
wff CoMembEr 𝐴
Syntax	cfunss 38261	Extend the definition of a class to include the function set class.
class Funss
Syntax	cfunsALTV 38262	Extend the definition of a class to include the functions class, i.e., the function relations class.
class FunsALTV
Syntax	wfunALTV 38263	Extend the definition of a wff to include the function predicate, i.e., the function relation predicate. (Read: 𝐹 is a function.)
wff FunALTV 𝐹
Syntax	cdisjss 38264	Extend the definition of a class to include the disjoint set class.
class Disjss
Syntax	cdisjs 38265	Extend the definition of a class to include the disjoints class, i.e., the disjoint relations class.
class Disjs
Syntax	wdisjALTV 38266	Extend the definition of a wff to include the disjoint predicate, i.e., the disjoint relation predicate. (Read: 𝑅 is a disjoint.)
wff Disj 𝑅
Syntax	celdisjs 38267	Extend the definition of a class to include the disjoint elements class, i.e., the disjoint element relations class.
class ElDisjs
Syntax	weldisj 38268	Extend the definition of a wff to include the disjoint element predicate, i.e., the disjoint element relation predicate. (Read: the elements of 𝐴 are disjoint.)
wff ElDisj 𝐴
Syntax	wantisymrel 38269	Extend the definition of a wff to include the antisymmetry relation predicate. (Read: 𝑅 is an antisymmetric relation.)
wff AntisymRel 𝑅
Syntax	cparts 38270	Extend the definition of a class to include the partitions class, i.e., the partition relations class.
class Parts
Syntax	wpart 38271	Extend the definition of a wff to include the partition predicate, i.e., the partition relation predicate. (Read: 𝐴 is a partition by 𝑅.)
wff 𝑅 Part 𝐴
Syntax	cmembparts 38272	Extend the definition of a class to include the member partitions class, i.e., the member partition relations class.
class MembParts
Syntax	wmembpart 38273	Extend the definition of a wff to include the member partition predicate, i.e., the member partition relation predicate. (Read: 𝐴 is a member partition.)
wff MembPart 𝐴
21.26.2  Preparatory theorems
Theorem	el2v1 38274	New way (elv 3443, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 23-Oct-2018.)
⊢ ((𝑥 ∈ V ∧ 𝜑) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	el3v1 38275	New way (elv 3443, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.)
⊢ ((𝑥 ∈ V ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
Theorem	el3v2 38276	New way (elv 3443, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.)
⊢ ((𝜑 ∧ 𝑦 ∈ V ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
Theorem	el3v12 38277	New way (elv 3443, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜒 → 𝜃)
Theorem	el3v13 38278	New way (elv 3443, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)
⊢ ((𝑥 ∈ V ∧ 𝜓 ∧ 𝑧 ∈ V) → 𝜃)    ⇒   ⊢ (𝜓 → 𝜃)
Theorem	el3v23 38279	New way (elv 3443, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)
⊢ ((𝜑 ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	anan 38280	Multiple commutations in conjunction. (Contributed by Peter Mazsa, 7-Mar-2020.)
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ ((𝜑 ∧ 𝜃) ∧ 𝜏)) ↔ ((𝜓 ∧ 𝜃) ∧ (𝜑 ∧ (𝜒 ∧ 𝜏))))
Theorem	triantru3 38281	A wff is equivalent to its conjunctions with truths. (Contributed by Peter Mazsa, 30-Nov-2018.)
⊢ 𝜑    &   ⊢ 𝜓    ⇒   ⊢ (𝜒 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	biorfd 38282	A wff is equivalent to its disjunction with falsehood, deduction form. (Contributed by Peter Mazsa, 22-Aug-2023.)
⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → (𝜒 ↔ (𝜓 ∨ 𝜒)))
Theorem	eqbrtr 38283	Substitution of equal classes in binary relation. (Contributed by Peter Mazsa, 14-Jun-2024.)
⊢ ((𝐴 = 𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Theorem	eqbrb 38284	Substitution of equal classes in a binary relation. (Contributed by Peter Mazsa, 14-Jun-2024.)
⊢ ((𝐴 = 𝐵 ∧ 𝐴𝑅𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵𝑅𝐶))
Theorem	eqeltr 38285	Substitution of equal classes into element relation. (Contributed by Peter Mazsa, 22-Jul-2017.)
⊢ ((𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)
Theorem	eqelb 38286	Substitution of equal classes into element relation. (Contributed by Peter Mazsa, 17-Jul-2019.)
⊢ ((𝐴 = 𝐵 ∧ 𝐴 ∈ 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶))
Theorem	eqeqan2d 38287	Implication of introducing a new equality. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
Theorem	disjresin 38288	The restriction to a disjoint is the empty class. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝑅 ↾ (𝐴 ∩ 𝐵)) = ∅)
Theorem	disjresdisj 38289	The intersection of restrictions to disjoint is the empty class. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∩ (𝑅 ↾ 𝐵)) = ∅)
Theorem	disjresdif 38290	The difference between restrictions to disjoint is the first restriction. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))
Theorem	disjresundif 38291	Lemma for ressucdifsn2 38509. (Contributed by Peter Mazsa, 24-Jul-2024.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))
Theorem	inres2 38292	Two ways of expressing the restriction of an intersection. (Contributed by Peter Mazsa, 5-Jun-2021.)
⊢ ((𝑅 ↾ 𝐴) ∩ 𝑆) = ((𝑅 ∩ 𝑆) ↾ 𝐴)
Theorem	coideq 38293	Equality theorem for composition of two classes. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐴) = (𝐵 ∘ 𝐵))
Theorem	nexmo1 38294	If there is no case where wff is true, it is true for at most one case. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
Theorem	eqab2 38295	Implication of a class abstraction. (Contributed by Peter Mazsa, 16-Apr-2019.)
⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑) → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	r2alan 38296*	Double restricted universal quantification, special case. (Contributed by Peter Mazsa, 17-Jun-2020.)
⊢ (∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) → 𝜓) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓))
Theorem	ssrabi 38297	Inference of restricted abstraction subclass from implication. (Contributed by Peter Mazsa, 26-Oct-2022.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓}
Theorem	rabimbieq 38298	Restricted equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 22-Jul-2021.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}    &   ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}
Theorem	abeqin 38299*	Intersection with class abstraction. (Contributed by Peter Mazsa, 21-Jul-2021.)
⊢ 𝐴 = (𝐵 ∩ 𝐶)    &   ⊢ 𝐵 = {𝑥 ∣ 𝜑}    ⇒   ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜑}
Theorem	abeqinbi 38300*	Intersection with class abstraction and equivalent wff's. (Contributed by Peter Mazsa, 21-Jul-2021.)
⊢ 𝐴 = (𝐵 ∩ 𝐶)    &   ⊢ 𝐵 = {𝑥 ∣ 𝜑}    &   ⊢ (𝑥 ∈ 𝐶 → (𝜑 ↔ 𝜓))    ⇒   ⊢ 𝐴 = {𝑥 ∈ 𝐶 ∣ 𝜓}