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Theorem List for Metamath Proof Explorer - 35201-35300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremismaxidl 35201* 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 35202 A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)
((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅))
 
Theoremmaxidlnr 35203 A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀𝑋)
 
Theoremmaxidlmax 35204 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 35205 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 35206 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)
 
20.20.21  Prime rings and integral domains
 
Syntaxcprrng 35207 Extend class notation with the class of prime rings.
class PrRing
 
Syntaxcdmn 35208 Extend class notation with the class of domains.
class Dmn
 
Definitiondf-prrngo 35209 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 35210 Define the class of (integral) domains. A domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
Dmn = (PrRing ∩ Com2)
 
Theoremisprrngo 35211 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 35212 A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑅 ∈ PrRing → 𝑅 ∈ RingOps)
 
Theoremsmprngopr 35213 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 35214 A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
(𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing)
 
Theoremisdmn 35215 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))
 
Theoremisdmn2 35216 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
 
Theoremdmncrng 35217 A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
(𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)
 
Theoremdmnrngo 35218 A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
(𝑅 ∈ Dmn → 𝑅 ∈ RingOps)
 
Theoremflddmn 35219 A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝐾 ∈ Fld → 𝐾 ∈ Dmn)
 
20.20.22  Ideal generators
 
Syntaxcigen 35220 Extend class notation with the ideal generation function.
class IdlGen
 
Definitiondf-igen 35221* 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 35222* 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 35223 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 35224 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 35225 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 35226 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 35227* 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 35228* 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 35229 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 35230 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 35231* 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 35232 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 35233 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 35234 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 35235* 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 35236 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 35237 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 35238 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))
 
20.21  Mathbox for Giovanni Mascellani
 
20.21.1  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 35239 A proof by contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
𝜑 → ⊥)       𝜑
 
Theoremnotbinot1 35240 Simplification rule of negation across a biimplication. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(¬ (¬ 𝜑𝜓) ↔ (𝜑𝜓))
 
Theorembicontr 35241 Biimplication of its own negation is a contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
((¬ 𝜑𝜑) ↔ ⊥)
 
Theoremimpor 35242 An equivalent formula for implying a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
((𝜑 → (𝜓𝜒)) ↔ ((¬ 𝜑𝜓) ∨ 𝜒))
 
Theoremorfa 35243 The falsum can be removed from a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
((𝜑 ∨ ⊥) ↔ 𝜑)
 
Theoremnotbinot2 35244 Commutation rule between negation and biimplication. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(¬ (𝜑𝜓) ↔ (¬ 𝜑𝜓))
 
Theorembiimpor 35245 A rewriting rule for biimplication. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(((𝜑𝜓) → 𝜒) ↔ ((¬ 𝜑𝜓) ∨ 𝜒))
 
Theoremorfa1 35246 Add a contradicting disjunct to an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑𝜓)       ((𝜑 ∨ ⊥) → 𝜓)
 
Theoremorfa2 35247 Remove a contradicting disjunct from an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → ⊥)       ((𝜑𝜓) → 𝜓)
 
Theorembifald 35248 Infer the equivalence to a contradiction from a negation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → ¬ 𝜓)       (𝜑 → (𝜓 ↔ ⊥))
 
Theoremorsild 35249 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → ¬ (𝜓𝜒))       (𝜑 → ¬ 𝜓)
 
Theoremorsird 35250 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → ¬ (𝜓𝜒))       (𝜑 → ¬ 𝜒)
 
Theoremcnf1dd 35251 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 35252 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 35253 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 35254 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 35255 A rearrangement of disjuncts, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → (𝜓 → ((𝜒𝜃) ∨ 𝜏)))       (𝜑 → (𝜓 → ((𝜒𝜏) ∨ 𝜃)))
 
Theoremnotornotel1 35256 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → ¬ (¬ 𝜓𝜒))       (𝜑𝜓)
 
Theoremnotornotel2 35257 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → ¬ (𝜓 ∨ ¬ 𝜒))       (𝜑𝜒)
 
Theoremcontrd 35258 A proof by contradiction, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → (¬ 𝜓𝜒))    &   (𝜑 → (¬ 𝜓 → ¬ 𝜒))       (𝜑𝜓)
 
Theoreman12i 35259 An inference from commuting operands in a chain of conjunctions. (Contributed by Giovanni Mascellani, 22-May-2019.)
(𝜑 ∧ (𝜓𝜒))       (𝜓 ∧ (𝜑𝜒))
 
Theoremexmid2 35260 An excluded middle law. (Contributed by Giovanni Mascellani, 23-May-2019.)
((𝜓𝜑) → 𝜒)    &   ((¬ 𝜓𝜂) → 𝜒)       ((𝜑𝜂) → 𝜒)
 
Theoremselconj 35261 An inference for selecting one of a list of conjuncts. (Contributed by Giovanni Mascellani, 23-May-2019.)
(𝜑 ↔ (𝜓𝜒))       ((𝜂𝜑) ↔ (𝜓 ∧ (𝜂𝜒)))
 
Theoremtruconj 35262 Add true as a conjunct. (Contributed by Giovanni Mascellani, 23-May-2019.)
(𝜑 ↔ (⊤ ∧ 𝜑))
 
Theoremorel 35263 An inference for disjunction elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
((𝜓𝜂) → 𝜃)    &   ((𝜒𝜌) → 𝜃)    &   (𝜑 → (𝜓𝜒))       ((𝜑 ∧ (𝜂𝜌)) → 𝜃)
 
Theoremnegel 35264 An inference for negation elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
(𝜓𝜒)    &   (𝜑 → ¬ 𝜒)       ((𝜑𝜓) → ⊥)
 
Theorembotel 35265 An inference for bottom elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
(𝜑 → ⊥)       (𝜑𝜓)
 
Theoremtradd 35266 Add top ad a conjunct. (Contributed by Giovanni Mascellani, 24-May-2019.)
(𝜑𝜓)       (𝜑 ↔ (⊤ ∧ 𝜓))
 
Theoremgm-sbtru 35267 Substitution does not change truth. (Contributed by Giovanni Mascellani, 24-May-2019.)
𝐴 ∈ V       ([𝐴 / 𝑥]⊤ ↔ ⊤)
 
Theoremsbfal 35268 Substitution does not change falsity. (Contributed by Giovanni Mascellani, 24-May-2019.)
𝐴 ∈ V       ([𝐴 / 𝑥]⊥ ↔ ⊥)
 
Theoremsbcani 35269 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 35270 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 35271 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))
 
Theoremsbcni 35272 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 35273 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 35274 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 35275* 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 35276* 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 35277* 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 35278* 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)
 
Theoremspsbcdi 35279 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 35280* 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 35281 A lemma for introducing an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
(𝜑𝜓)    &   ([𝐴 / 𝑥]𝜒𝜓)       (𝜑 → ∃𝑥𝜒)
 
Theoremexlimddvf 35282 A lemma for eliminating an existential quantifier. (Contributed by Giovanni Mascellani, 30-May-2019.)
(𝜑 → ∃𝑥𝜃)    &   𝑥𝜓    &   ((𝜃𝜓) → 𝜒)    &   𝑥𝜒       ((𝜑𝜓) → 𝜒)
 
Theoremexlimddvfi 35283 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 35284 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 35285* Lemma for sbccom2 35286. (Contributed by Giovanni Mascellani, 31-May-2019.)
𝐴 ∈ V       ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
 
Theoremsbccom2 35286* Commutative law for double class substitution. (Contributed by Giovanni Mascellani, 31-May-2019.)
𝐴 ∈ V       ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
 
Theoremsbccom2f 35287* Commutative law for double class substitution, with nonfree variable condition. (Contributed by Giovanni Mascellani, 31-May-2019.)
𝐴 ∈ V    &   𝑦𝐴       ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
 
Theoremsbccom2fi 35288* Commutative law for double class substitution, with nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.)
𝐴 ∈ V    &   𝑦𝐴    &   𝐴 / 𝑥𝐵 = 𝐶    &   ([𝐴 / 𝑥]𝜑𝜓)       ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜓)
 
Theoremcsbcom2fi 35289* Commutative law for double class substitution in a class, with nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.)
𝐴 ∈ V    &   𝑦𝐴    &   𝐴 / 𝑥𝐵 = 𝐶    &   𝐴 / 𝑥𝐷 = 𝐸       𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐸
 
20.21.2  Tseitin axioms

A collection of Tseitin axioms used to convert a wff to Conjunctive Normal Form.

 
Theoremfald 35290 Refutation of falsity, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ¬ ⊥)
 
Theoremtsim1 35291 A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))
 
Theoremtsim2 35292 A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜑 ∨ (𝜑𝜓)))
 
Theoremtsim3 35293 A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (¬ 𝜓 ∨ (𝜑𝜓)))
 
Theoremtsbi1 35294 A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
 
Theoremtsbi2 35295 A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((𝜑𝜓) ∨ (𝜑𝜓)))
 
Theoremtsbi3 35296 A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))
 
Theoremtsbi4 35297 A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))
 
Theoremtsxo1 35298 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))
 
Theoremtsxo2 35299 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
 
Theoremtsxo3 35300 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
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