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Theorem List for Metamath Proof Explorer - 34001-34100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempridlc2 34001 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 34002 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 34003* 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 34004 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 34005 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 34006 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 34007 A proof by contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
𝜑 → ⊥)       𝜑

Theoremnotbinot1 34008 Simplification rule of negation across a biimplication. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(¬ (¬ 𝜑𝜓) ↔ (𝜑𝜓))

Theorembicontr 34009 Biimplication of its own negation is a contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
((¬ 𝜑𝜑) ↔ ⊥)

Theoremimpor 34010 An equivalent formula for implying a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
((𝜑 → (𝜓𝜒)) ↔ ((¬ 𝜑𝜓) ∨ 𝜒))

Theoremorfa 34011 The falsum can be removed from a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
((𝜑 ∨ ⊥) ↔ 𝜑)

Theoremnotbinot2 34012 Commutation rule between negation and biimplication. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(¬ (𝜑𝜓) ↔ (¬ 𝜑𝜓))

Theorembiimpor 34013 A rewriting rule for biimplication. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(((𝜑𝜓) → 𝜒) ↔ ((¬ 𝜑𝜓) ∨ 𝜒))

Theoremunitresl 34014 A lemma for Conjunctive Normal Form unit propagation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → ¬ 𝜒)       (𝜑𝜓)

Theoremunitresr 34015 A lemma for Conjunctive Normal Form unit propagation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → ¬ 𝜓)       (𝜑𝜒)

Theoremorfa1 34016 Add a contradicting disjunct to an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑𝜓)       ((𝜑 ∨ ⊥) → 𝜓)

Theoremorfa2 34017 Remove a contradicting disjunct from an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → ⊥)       ((𝜑𝜓) → 𝜓)

Theorembifald 34018 Infer the equivalence to a contradiction from a negation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → ¬ 𝜓)       (𝜑 → (𝜓 ↔ ⊥))

Theoremorsild 34019 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → ¬ (𝜓𝜒))       (𝜑 → ¬ 𝜓)

Theoremorsird 34020 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(𝜑 → ¬ (𝜓𝜒))       (𝜑 → ¬ 𝜒)

Theoremorcomdd 34021 Commutativity of logic disjunction, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓 → (𝜃𝜒)))

Theoremcnf1dd 34022 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 34023 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 34024 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 34025 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 34026 A rearrangement of disjuncts, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → (𝜓 → ((𝜒𝜃) ∨ 𝜏)))       (𝜑 → (𝜓 → ((𝜒𝜏) ∨ 𝜃)))

Theoremnotornotel1 34027 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → ¬ (¬ 𝜓𝜒))       (𝜑𝜓)

Theoremnotornotel2 34028 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → ¬ (𝜓 ∨ ¬ 𝜒))       (𝜑𝜒)

Theoremcontrd 34029 A proof by contradiction, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(𝜑 → (¬ 𝜓𝜒))    &   (𝜑 → (¬ 𝜓 → ¬ 𝜒))       (𝜑𝜓)

Theoreman12i 34030 An inference from commuting operands in a chain of conjunctions. (Contributed by Giovanni Mascellani, 22-May-2019.)
(𝜑 ∧ (𝜓𝜒))       (𝜓 ∧ (𝜑𝜒))

Theoremexmid2 34031 An excluded middle law. (Contributed by Giovanni Mascellani, 23-May-2019.)
((𝜓𝜑) → 𝜒)    &   ((¬ 𝜓𝜂) → 𝜒)       ((𝜑𝜂) → 𝜒)

Theoremselconj 34032 An inference for selecting one of a list of conjuncts. (Contributed by Giovanni Mascellani, 23-May-2019.)
(𝜑 ↔ (𝜓𝜒))       ((𝜂𝜑) ↔ (𝜓 ∧ (𝜂𝜒)))

Theoremtruconj 34033 Add true as a conjunct. (Contributed by Giovanni Mascellani, 23-May-2019.)
(𝜑 ↔ (⊤ ∧ 𝜑))

Theoremorel 34034 An inference for disjunction elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
((𝜓𝜂) → 𝜃)    &   ((𝜒𝜌) → 𝜃)    &   (𝜑 → (𝜓𝜒))       ((𝜑 ∧ (𝜂𝜌)) → 𝜃)

Theoremnegel 34035 An inference for negation elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
(𝜓𝜒)    &   (𝜑 → ¬ 𝜒)       ((𝜑𝜓) → ⊥)

Theorembotel 34036 An inference for bottom elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
(𝜑 → ⊥)       (𝜑𝜓)

(𝜑𝜓)       (𝜑 ↔ (⊤ ∧ 𝜓))

Theoremsbtru 34038 Substitution does not change truth. (Contributed by Giovanni Mascellani, 24-May-2019.)
𝐴 ∈ V       ([𝐴 / 𝑥]⊤ ↔ ⊤)

Theoremsbfal 34039 Substitution does not change falsity. (Contributed by Giovanni Mascellani, 24-May-2019.)
𝐴 ∈ V       ([𝐴 / 𝑥]⊥ ↔ ⊥)

Theoremsbcani 34040 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 34041 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 34042 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 34043 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 34044 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 34045 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 34046 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 34047* 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 34048* 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 34049* 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 34050* 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 34051 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 34052* 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 34053 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 34054* 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 34055 A lemma for introducing an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
(𝜑𝜓)    &   ([𝐴 / 𝑥]𝜒𝜓)       (𝜑 → ∃𝑥𝜒)

Theoremexlimddvf 34056 A lemma for eliminating an existential quantifier. (Contributed by Giovanni Mascellani, 30-May-2019.)
(𝜑 → ∃𝑥𝜃)    &   𝑥𝜓    &   ((𝜃𝜓) → 𝜒)    &   𝑥𝜒       ((𝜑𝜓) → 𝜒)

Theoremexlimddvfi 34057 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 34058 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 34059* Lemma for sbccom2 34060. (Contributed by Giovanni Mascellani, 31-May-2019.)
𝐴 ∈ V       ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Theoremsbccom2 34060* 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 34061* Commutative law for double class substitution, with non free variable condition. (Contributed by Giovanni Mascellani, 31-May-2019.)
𝐴 ∈ V    &   𝑦𝐴       ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Theoremsbccom2fi 34062* Commutative law for double class substitution, with non free 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)

Theoremsbcgfi 34063 Substitution for a variable not free in a wff does not affect it, in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.)
𝐴 ∈ V    &   𝑥𝜑       ([𝐴 / 𝑥]𝜑𝜑)

Theoremcsbcom2fi 34064* Commutative law for double class substitution in a class, with non free 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

Theoremcsbgfi 34065 Substitution for a variable not free in a class does not affect it, in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.)
𝐴 ∈ V    &   𝑥𝐵       𝐴 / 𝑥𝐵 = 𝐵

20.20.2  Tseitin axioms

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

Theoremfald 34066 Refutation of falsity, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ¬ ⊥)

Theoremtsim1 34067 A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsim2 34068 A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜑 ∨ (𝜑𝜓)))

Theoremtsim3 34069 A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (¬ 𝜓 ∨ (𝜑𝜓)))

Theoremtsbi1 34070 A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))

Theoremtsbi2 34071 A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((𝜑𝜓) ∨ (𝜑𝜓)))

Theoremtsbi3 34072 A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsbi4 34073 A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsxo1 34074 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsxo2 34075 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsxo3 34076 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))

Theoremtsxo4 34077 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑𝜓) ∨ (𝜑𝜓)))

Theoremtsan1 34078 A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))

Theoremtsan2 34079 A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜑 ∨ ¬ (𝜑𝜓)))

Theoremtsan3 34080 A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜓 ∨ ¬ (𝜑𝜓)))

Theoremtsna1 34081 A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsna2 34082 A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜑 ∨ (𝜑𝜓)))

Theoremtsna3 34083 A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜓 ∨ (𝜑𝜓)))

Theoremtsor1 34084 A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsor2 34085 A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → (¬ 𝜑 ∨ (𝜑𝜓)))

Theoremtsor3 34086 A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → (¬ 𝜓 ∨ (𝜑𝜓)))

Theoremts3an1 34087 A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → ((¬ (𝜑𝜓) ∨ ¬ 𝜒) ∨ (𝜑𝜓𝜒)))

Theoremts3an2 34088 A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → ((𝜑𝜓) ∨ ¬ (𝜑𝜓𝜒)))

Theoremts3an3 34089 A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → (𝜒 ∨ ¬ (𝜑𝜓𝜒)))

Theoremts3or1 34090 A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → (((𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑𝜓𝜒)))

Theoremts3or2 34091 A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → (¬ (𝜑𝜓) ∨ (𝜑𝜓𝜒)))

Theoremts3or3 34092 A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → (¬ 𝜒 ∨ (𝜑𝜓𝜒)))

20.20.3  Equality deductions

A collection of theorems for commuting equalities (or biimplications) with other constructs.

Theoremiuneq2f 34093 Equality deduction for indexed union. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theoremabeq12 34094 Equality deduction for class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
(∀𝑥(𝜑𝜓) → {𝑥𝜑} = {𝑥𝜓})

Theoremrabeq12f 34095 Equality deduction for restricted class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
𝑥𝐴    &   𝑥𝐵       ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) → {𝑥𝐴𝜑} = {𝑥𝐵𝜓})

Theoremcsbeq12 34096 Equality deduction for substitution in class. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → 𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐷)

Theoremnfbii2 34097 Equality deduction for not-freeness. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
(∀𝑥(𝜑𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))

Theoremsbeqi 34098 Equality deduction for substitution. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
((𝑥 = 𝑦 ∧ ∀𝑧(𝜑𝜓)) → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜓))

Theoremralbi12f 34099 Equality deduction for restricted universal quantification. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
𝑥𝐴    &   𝑥𝐵       ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))

Theoremoprabbi 34100 Equality deduction for class abstraction of nested ordered pairs. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
(∀𝑥𝑦𝑧(𝜑𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓})

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