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Theorem List for Metamath Proof Explorer - 33801-33900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremunichnidl 33801* The union of a nonempty chain of ideals is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)
((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → 𝐶 ∈ (Idl‘𝑅))

Theoremkeridl 33802 The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.)
𝐺 = (1st𝑆)    &   𝑍 = (GId‘𝐺)       ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹 “ {𝑍}) ∈ (Idl‘𝑅))

Theorempridlval 33803* The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ RingOps → (PrIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})

Theoremispridl 33804* The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))

Theorempridlidl 33805 A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ∈ (Idl‘𝑅))

Theorempridlnr 33806 A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃𝑋)

Theorempridl 33807* The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐻 = (2nd𝑅)       (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (Idl‘𝑅) ∧ 𝐵 ∈ (Idl‘𝑅) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃)) → (𝐴𝑃𝐵𝑃))

Theoremispridl2 33808* A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 33840 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))) → 𝑃 ∈ (PrIdl‘𝑅))

Theoremmaxidlval 33809* The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ RingOps → (MaxIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝑋)))})

Theoremismaxidl 33810* The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)))))

Theoremmaxidlidl 33811 A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)
((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))

Theoremmaxidlnr 33812 A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀𝑋)

Theoremmaxidlmax 33813 A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑀𝐼)) → (𝐼 = 𝑀𝐼 = 𝑋))

Theoremmaxidln1 33814 One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.)
𝐻 = (2nd𝑅)    &   𝑈 = (GId‘𝐻)       ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈𝑀)

Theoremmaxidln0 33815 A ring with a maximal ideal is not the zero ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑍 = (GId‘𝐺)    &   𝑈 = (GId‘𝐻)       ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈𝑍)

20.19.21  Prime rings and integral domains

Syntaxcprrng 33816 Extend class notation with the class of prime rings.
class PrRing

Syntaxcdmn 33817 Extend class notation with the class of domains.
class Dmn

Definitiondf-prrngo 33818 Define the class of prime rings. A ring is prime if the zero ideal is a prime ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟)}

Definitiondf-dmn 33819 Define the class of (integral) domains. A domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
Dmn = (PrRing ∩ Com2)

Theoremisprrngo 33820 The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑍 = (GId‘𝐺)       (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))

Theoremprrngorngo 33821 A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑅 ∈ PrRing → 𝑅 ∈ RingOps)

Theoremsmprngopr 33822 A simple ring (one whose only ideals are 0 and 𝑅) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)    &   𝑈 = (GId‘𝐻)       ((𝑅 ∈ RingOps ∧ 𝑈𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → 𝑅 ∈ PrRing)

Theoremdivrngpr 33823 A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
(𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing)

Theoremisdmn 33824 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))

Theoremisdmn2 33825 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))

Theoremdmncrng 33826 A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
(𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)

Theoremdmnrngo 33827 A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
(𝑅 ∈ Dmn → 𝑅 ∈ RingOps)

Theoremflddmn 33828 A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝐾 ∈ Fld → 𝐾 ∈ Dmn)

20.19.22  Ideal generators

Syntaxcigen 33829 Extend class notation with the ideal generation function.
class IdlGen

Definitiondf-igen 33830* Define the ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st𝑟) ↦ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠𝑗})

Theoremigenval 33831* The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) = {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆𝑗})

Theoremigenss 33832 A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))

Theoremigenidl 33833 The ideal generated by a set is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))

Theoremigenmin 33834 The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.)
((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)

Theoremigenidl2 33835 The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = 𝐼)

Theoremigenval2 33836* The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝑆𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 ↔ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆𝑗𝐼𝑗))))

Theoremprnc 33837* A principal ideal (an ideal generated by one element) in a commutative ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ CRingOps ∧ 𝐴𝑋) → (𝑅 IdlGen {𝐴}) = {𝑥𝑋 ∣ ∃𝑦𝑋 𝑥 = (𝑦𝐻𝐴)})

Theoremisfldidl 33838 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝐾)    &   𝐻 = (2nd𝐾)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)    &   𝑈 = (GId‘𝐻)       (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))

Theoremisfldidl2 33839 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.)
𝐺 = (1st𝐾)    &   𝐻 = (2nd𝐾)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))

Theoremispridlc 33840* The predicate "is a prime ideal". Alternate definition for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))

Theorempridlc 33841 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴𝑃𝐵𝑃))

Theorempridlc2 33842 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → 𝐵𝑃)

Theorempridlc3 33843 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋𝑃) ∧ 𝐵 ∈ (𝑋𝑃))) → (𝐴𝐻𝐵) ∈ (𝑋𝑃))

Theoremisdmn3 33844* The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)    &   𝑈 = (GId‘𝐻)       (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ 𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍))))

Theoremdmnnzd 33845 A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐻𝐵) = 𝑍)) → (𝐴 = 𝑍𝐵 = 𝑍))

Theoremdmncan1 33846 Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) → 𝐵 = 𝐶))

Theoremdmncan2 33847 Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐶𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) → 𝐴 = 𝐵))

20.20  Mathbox for Giovanni Mascellani

20.20.1  Tools for automatic proof building

The results in this section are mostly meant for being used by automatic proof building programs. As a result, they might appear less useful or meaningful than others to human beings.

Theoremefald2 33848 A proof by contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
𝜑 → ⊥)       𝜑

Theoremnotbinot1 33849 Simplification rule of negation across a biimplication. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(¬ (¬ 𝜑𝜓) ↔ (𝜑𝜓))

Theorembicontr 33850 Biimplication of its own negation is a contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
((¬ 𝜑𝜑) ↔ ⊥)

Theoremimpor 33851 An equivalent formula for implying a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
((𝜑 → (𝜓𝜒)) ↔ ((¬ 𝜑𝜓) ∨ 𝜒))

Theoremorfa 33852 The falsum can be removed from a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
((𝜑 ∨ ⊥) ↔ 𝜑)

Theoremnotbinot2 33853 Commutation rule between negation and biimplication. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(¬ (𝜑𝜓) ↔ (¬ 𝜑𝜓))

Theorembiimpor 33854 A rewriting rule for biimplication. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(((𝜑𝜓) → 𝜒) ↔ ((¬ 𝜑𝜓) ∨ 𝜒))

Theoremunitresl 33855 A lemma for Conjunctive Normal Form unit propagation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → ¬ 𝜒)       (𝜑𝜓)

Theoremunitresr 33856 A lemma for Conjunctive Normal Form unit propagation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → ¬ 𝜓)       (𝜑𝜒)

Theoremorfa1 33857 Add a contradicting disjunct to an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑𝜓)       ((𝜑 ∨ ⊥) → 𝜓)

Theoremorfa2 33858 Remove a contradicting disjunct from an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → ⊥)       ((𝜑𝜓) → 𝜓)

Theorembifald 33859 Infer the equivalence to a contradiction from a negation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → ¬ 𝜓)       (𝜑 → (𝜓 ↔ ⊥))

Theoremorsild 33860 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → ¬ (𝜓𝜒))       (𝜑 → ¬ 𝜓)

Theoremorsird 33861 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → ¬ (𝜓𝜒))       (𝜑 → ¬ 𝜒)

Theoremorcomdd 33862 Commutativity of logic disjunction, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓 → (𝜃𝜒)))

Theoremcnf1dd 33863 A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → (𝜓 → ¬ 𝜒))    &   (𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓𝜃))

Theoremcnf2dd 33864 A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → (𝜓 → ¬ 𝜃))    &   (𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓𝜒))

Theoremcnfn1dd 33865 A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓 → (¬ 𝜒𝜃)))       (𝜑 → (𝜓𝜃))

Theoremcnfn2dd 33866 A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜓 → (𝜒 ∨ ¬ 𝜃)))       (𝜑 → (𝜓𝜒))

Theoremor32dd 33867 A rearrangement of disjuncts, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → (𝜓 → ((𝜒𝜃) ∨ 𝜏)))       (𝜑 → (𝜓 → ((𝜒𝜏) ∨ 𝜃)))

Theoremnotornotel1 33868 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → ¬ (¬ 𝜓𝜒))       (𝜑𝜓)

Theoremnotornotel2 33869 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → ¬ (𝜓 ∨ ¬ 𝜒))       (𝜑𝜒)

Theoremcontrd 33870 A proof by contradiction, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → (¬ 𝜓𝜒))    &   (𝜑 → (¬ 𝜓 → ¬ 𝜒))       (𝜑𝜓)

Theoreman12i 33871 An inference from commuting operands in a chain of conjunctions. (Contributed by Giovanni Mascellani, 22-May-2019.)
(𝜑 ∧ (𝜓𝜒))       (𝜓 ∧ (𝜑𝜒))

Theoremexmid2 33872 An excluded middle law. (Contributed by Giovanni Mascellani, 23-May-2019.)
((𝜓𝜑) → 𝜒)    &   ((¬ 𝜓𝜂) → 𝜒)       ((𝜑𝜂) → 𝜒)

Theoremselconj 33873 An inference for selecting one of a list of conjuncts. (Contributed by Giovanni Mascellani, 23-May-2019.)
(𝜑 ↔ (𝜓𝜒))       ((𝜂𝜑) ↔ (𝜓 ∧ (𝜂𝜒)))

Theoremtruconj 33874 Add true as a conjunct. (Contributed by Giovanni Mascellani, 23-May-2019.)
(𝜑 ↔ (⊤ ∧ 𝜑))

Theoremorel 33875 An inference for disjunction elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
((𝜓𝜂) → 𝜃)    &   ((𝜒𝜌) → 𝜃)    &   (𝜑 → (𝜓𝜒))       ((𝜑 ∧ (𝜂𝜌)) → 𝜃)

Theoremnegel 33876 An inference for negation elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
(𝜓𝜒)    &   (𝜑 → ¬ 𝜒)       ((𝜑𝜓) → ⊥)

Theorembotel 33877 An inference for bottom elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
(𝜑 → ⊥)       (𝜑𝜓)

(𝜑𝜓)       (𝜑 ↔ (⊤ ∧ 𝜓))

Theoremsbtru 33879 Substitution does not change truth. (Contributed by Giovanni Mascellani, 24-May-2019.)
𝐴 ∈ V       ([𝐴 / 𝑥]⊤ ↔ ⊤)

Theoremsbfal 33880 Substitution does not change falsity. (Contributed by Giovanni Mascellani, 24-May-2019.)
𝐴 ∈ V       ([𝐴 / 𝑥]⊥ ↔ ⊥)

Theoremsbcani 33881 Distribution of class substitution over conjunction, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
([𝐴 / 𝑥]𝜑𝜒)    &   ([𝐴 / 𝑥]𝜓𝜂)       ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜒𝜂))

Theoremsbcori 33882 Distribution of class substitution over disjunction, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
([𝐴 / 𝑥]𝜑𝜒)    &   ([𝐴 / 𝑥]𝜓𝜂)       ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜒𝜂))

Theoremsbcimi 33883 Distribution of class substitution over implication, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
𝐴 ∈ V    &   ([𝐴 / 𝑥]𝜑𝜒)    &   ([𝐴 / 𝑥]𝜓𝜂)       ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜒𝜂))

Theoremsbceqi 33884 Distribution of class substitution over equality, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
𝐴 ∈ V    &   𝐴 / 𝑥𝐵 = 𝐷    &   𝐴 / 𝑥𝐶 = 𝐸       ([𝐴 / 𝑥]𝐵 = 𝐶𝐷 = 𝐸)

Theoremsbcni 33885 Move class substitution inside a negation, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
𝐴 ∈ V    &   ([𝐴 / 𝑥]𝜑𝜓)       ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜓)

Theoremsbali 33886 Discard class substitution in a universal quantification when substituting the quantified variable, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
𝐴 ∈ V       ([𝐴 / 𝑥]𝑥𝜑 ↔ ∀𝑥𝜑)

Theoremsbexi 33887 Discard class substitution in an existential quantification when substituting the quantified variable, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
𝐴 ∈ V       ([𝐴 / 𝑥]𝑥𝜑 ↔ ∃𝑥𝜑)

Theoremsbcalf 33888* Move universal quantifier in and out of class substitution, with an explicit non-free variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.)
𝑦𝐴       ([𝐴 / 𝑥]𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)

Theoremsbcexf 33889* Move existential quantifier in and out of class substitution, with an explicit non-free variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.)
𝑦𝐴       ([𝐴 / 𝑥]𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)

Theoremsbcalfi 33890* Move universal quantifier in and out of class substitution, with an explicit non-free variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
𝑦𝐴    &   ([𝐴 / 𝑥]𝜑𝜓)       ([𝐴 / 𝑥]𝑦𝜑 ↔ ∀𝑦𝜓)

Theoremsbcexfi 33891* Move existential quantifier in and out of class substitution, with an explicit non-free variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
𝑦𝐴    &   ([𝐴 / 𝑥]𝜑𝜓)       ([𝐴 / 𝑥]𝑦𝜑 ↔ ∃𝑦𝜓)

Theoremcsbvargi 33892 The proper substitution of a class for a variable in that variable results in the class (if the class exists), in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
𝐴 ∈ V       𝐴 / 𝑥𝑥 = 𝐴

Theoremcsbconstgi 33893* The proper substitution of a class for a variable in another variable does not modify it, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
𝐴 ∈ V       𝐴 / 𝑥𝑦 = 𝑦

Theoremspsbcdi 33894 A lemma for eliminating a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
𝐴 ∈ V    &   (𝜑 → ∀𝑥𝜒)    &   ([𝐴 / 𝑥]𝜒𝜓)       (𝜑𝜓)

Theoremalrimii 33895* A lemma for introducing a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
𝑦𝜑    &   (𝜑𝜓)    &   ([𝑦 / 𝑥]𝜒𝜓)    &   𝑦𝜒       (𝜑 → ∀𝑥𝜒)

Theoremspesbcdi 33896 A lemma for introducing an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
(𝜑𝜓)    &   ([𝐴 / 𝑥]𝜒𝜓)       (𝜑 → ∃𝑥𝜒)

Theoremexlimddvf 33897 A lemma for eliminating an existential quantifier. (Contributed by Giovanni Mascellani, 30-May-2019.)
(𝜑 → ∃𝑥𝜃)    &   𝑥𝜓    &   ((𝜃𝜓) → 𝜒)    &   𝑥𝜒       ((𝜑𝜓) → 𝜒)

Theoremexlimddvfi 33898 A lemma for eliminating an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.)
(𝜑 → ∃𝑥𝜃)    &   𝑦𝜃    &   𝑦𝜓    &   ([𝑦 / 𝑥]𝜃𝜂)    &   ((𝜂𝜓) → 𝜒)    &   𝑦𝜒       ((𝜑𝜓) → 𝜒)

Theoremsbceq1ddi 33899 A lemma for eliminating inequality, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝜃)    &   ([𝐴 / 𝑥]𝜒𝜃)    &   ([𝐵 / 𝑥]𝜒𝜂)       ((𝜑𝜓) → 𝜂)

Theoremsbccom2lem 33900* Lemma for sbccom2 33901. (Contributed by Giovanni Mascellani, 31-May-2019.)
𝐴 ∈ V       ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

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