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Theorem List for Metamath Proof Explorer - 35801-35900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcdmn 35801 Extend class notation with the class of domains.
class Dmn

Definitiondf-prrngo 35802 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 35803 Define the class of (integral) domains. A domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
Dmn = (PrRing ∩ Com2)

Theoremisprrngo 35804 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 35805 A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑅 ∈ PrRing → 𝑅 ∈ RingOps)

Theoremsmprngopr 35806 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 35807 A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
(𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing)

Theoremisdmn 35808 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))

Theoremisdmn2 35809 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))

Theoremdmncrng 35810 A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
(𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)

Theoremdmnrngo 35811 A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
(𝑅 ∈ Dmn → 𝑅 ∈ RingOps)

Theoremflddmn 35812 A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝐾 ∈ Fld → 𝐾 ∈ Dmn)

20.20.22  Ideal generators

Syntaxcigen 35813 Extend class notation with the ideal generation function.
class IdlGen

Definitiondf-igen 35814* 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 35815* 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 35816 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 35817 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 35818 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 35819 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 35820* 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 35821* 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 35822 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 35823 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 35824* 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 35825 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 35826 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 35827 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 35828* 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 35829 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 35830 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 35831 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 35832 A proof by contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
𝜑 → ⊥)       𝜑

Theoremnotbinot1 35833 Simplification rule of negation across a biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(¬ (¬ 𝜑𝜓) ↔ (𝜑𝜓))

Theorembicontr 35834 Biconditional of its own negation is a contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
((¬ 𝜑𝜑) ↔ ⊥)

Theoremimpor 35835 An equivalent formula for implying a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
((𝜑 → (𝜓𝜒)) ↔ ((¬ 𝜑𝜓) ∨ 𝜒))

Theoremorfa 35836 The falsum can be removed from a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
((𝜑 ∨ ⊥) ↔ 𝜑)

Theoremnotbinot2 35837 Commutation rule between negation and biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(¬ (𝜑𝜓) ↔ (¬ 𝜑𝜓))

Theorembiimpor 35838 A rewriting rule for biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(((𝜑𝜓) → 𝜒) ↔ ((¬ 𝜑𝜓) ∨ 𝜒))

Theoremorfa1 35839 Add a contradicting disjunct to an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑𝜓)       ((𝜑 ∨ ⊥) → 𝜓)

Theoremorfa2 35840 Remove a contradicting disjunct from an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → ⊥)       ((𝜑𝜓) → 𝜓)

Theorembifald 35841 Infer the equivalence to a contradiction from a negation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → ¬ 𝜓)       (𝜑 → (𝜓 ↔ ⊥))

Theoremorsild 35842 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → ¬ (𝜓𝜒))       (𝜑 → ¬ 𝜓)

Theoremorsird 35843 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → ¬ (𝜓𝜒))       (𝜑 → ¬ 𝜒)

Theoremcnf1dd 35844 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 35845 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 35846 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 35847 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 35848 A rearrangement of disjuncts, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → (𝜓 → ((𝜒𝜃) ∨ 𝜏)))       (𝜑 → (𝜓 → ((𝜒𝜏) ∨ 𝜃)))

Theoremnotornotel1 35849 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → ¬ (¬ 𝜓𝜒))       (𝜑𝜓)

Theoremnotornotel2 35850 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → ¬ (𝜓 ∨ ¬ 𝜒))       (𝜑𝜒)

Theoremcontrd 35851 A proof by contradiction, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → (¬ 𝜓𝜒))    &   (𝜑 → (¬ 𝜓 → ¬ 𝜒))       (𝜑𝜓)

Theoreman12i 35852 An inference from commuting operands in a chain of conjunctions. (Contributed by Giovanni Mascellani, 22-May-2019.)
(𝜑 ∧ (𝜓𝜒))       (𝜓 ∧ (𝜑𝜒))

Theoremexmid2 35853 An excluded middle law. (Contributed by Giovanni Mascellani, 23-May-2019.)
((𝜓𝜑) → 𝜒)    &   ((¬ 𝜓𝜂) → 𝜒)       ((𝜑𝜂) → 𝜒)

Theoremselconj 35854 An inference for selecting one of a list of conjuncts. (Contributed by Giovanni Mascellani, 23-May-2019.)
(𝜑 ↔ (𝜓𝜒))       ((𝜂𝜑) ↔ (𝜓 ∧ (𝜂𝜒)))

Theoremtruconj 35855 Add true as a conjunct. (Contributed by Giovanni Mascellani, 23-May-2019.)
(𝜑 ↔ (⊤ ∧ 𝜑))

Theoremorel 35856 An inference for disjunction elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
((𝜓𝜂) → 𝜃)    &   ((𝜒𝜌) → 𝜃)    &   (𝜑 → (𝜓𝜒))       ((𝜑 ∧ (𝜂𝜌)) → 𝜃)

Theoremnegel 35857 An inference for negation elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
(𝜓𝜒)    &   (𝜑 → ¬ 𝜒)       ((𝜑𝜓) → ⊥)

Theorembotel 35858 An inference for bottom elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
(𝜑 → ⊥)       (𝜑𝜓)

(𝜑𝜓)       (𝜑 ↔ (⊤ ∧ 𝜓))

Theoremgm-sbtru 35860 Substitution does not change truth. (Contributed by Giovanni Mascellani, 24-May-2019.)
𝐴 ∈ V       ([𝐴 / 𝑥]⊤ ↔ ⊤)

Theoremsbfal 35861 Substitution does not change falsity. (Contributed by Giovanni Mascellani, 24-May-2019.)
𝐴 ∈ V       ([𝐴 / 𝑥]⊥ ↔ ⊥)

Theoremsbcani 35862 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 35863 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 35864 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 35865 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 35866 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 35867 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 35868* 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 35869* 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 35870* 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 35871* 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 35872 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 35873* 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 35874 A lemma for introducing an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
(𝜑𝜓)    &   ([𝐴 / 𝑥]𝜒𝜓)       (𝜑 → ∃𝑥𝜒)

Theoremexlimddvf 35875 A lemma for eliminating an existential quantifier. (Contributed by Giovanni Mascellani, 30-May-2019.)
(𝜑 → ∃𝑥𝜃)    &   𝑥𝜓    &   ((𝜃𝜓) → 𝜒)    &   𝑥𝜒       ((𝜑𝜓) → 𝜒)

Theoremexlimddvfi 35876 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 35877 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 35878* Lemma for sbccom2 35879. (Contributed by Giovanni Mascellani, 31-May-2019.)
𝐴 ∈ V       ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Theoremsbccom2 35879* 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 35880* 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 35881* 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 35882* 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 35883 Refutation of falsity, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ¬ ⊥)

Theoremtsim1 35884 A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsim2 35885 A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜑 ∨ (𝜑𝜓)))

Theoremtsim3 35886 A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (¬ 𝜓 ∨ (𝜑𝜓)))

Theoremtsbi1 35887 A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))

Theoremtsbi2 35888 A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((𝜑𝜓) ∨ (𝜑𝜓)))

Theoremtsbi3 35889 A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsbi4 35890 A Tseitin axiom for logical biconditional, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsxo1 35891 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsxo2 35892 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsxo3 35893 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))

Theoremtsxo4 35894 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑𝜓) ∨ (𝜑𝜓)))

Theoremtsan1 35895 A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))

Theoremtsan2 35896 A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜑 ∨ ¬ (𝜑𝜓)))

Theoremtsan3 35897 A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜓 ∨ ¬ (𝜑𝜓)))

Theoremtsna1 35898 A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsna2 35899 A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜑 ∨ (𝜑𝜓)))

Theoremtsna3 35900 A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜓 ∨ (𝜑𝜓)))

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