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Theorem List for Metamath Proof Explorer - 33901-34000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsbccom2 33901* 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 33902* 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 33903* 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 33904 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 33905* 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 33906 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 33907 Refutation of falsity, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ¬ ⊥)

Theoremtsim1 33908 A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsim2 33909 A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜑 ∨ (𝜑𝜓)))

Theoremtsim3 33910 A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (¬ 𝜓 ∨ (𝜑𝜓)))

Theoremtsbi1 33911 A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))

Theoremtsbi2 33912 A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((𝜑𝜓) ∨ (𝜑𝜓)))

Theoremtsbi3 33913 A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsbi4 33914 A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsxo1 33915 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsxo2 33916 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsxo3 33917 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))

Theoremtsxo4 33918 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑𝜓) ∨ (𝜑𝜓)))

Theoremtsan1 33919 A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))

Theoremtsan2 33920 A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜑 ∨ ¬ (𝜑𝜓)))

Theoremtsan3 33921 A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜓 ∨ ¬ (𝜑𝜓)))

Theoremtsna1 33922 A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsna2 33923 A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜑 ∨ (𝜑𝜓)))

Theoremtsna3 33924 A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜓 ∨ (𝜑𝜓)))

Theoremtsor1 33925 A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsor2 33926 A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → (¬ 𝜑 ∨ (𝜑𝜓)))

Theoremtsor3 33927 A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → (¬ 𝜓 ∨ (𝜑𝜓)))

Theoremts3an1 33928 A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → ((¬ (𝜑𝜓) ∨ ¬ 𝜒) ∨ (𝜑𝜓𝜒)))

Theoremts3an2 33929 A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → ((𝜑𝜓) ∨ ¬ (𝜑𝜓𝜒)))

Theoremts3an3 33930 A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → (𝜒 ∨ ¬ (𝜑𝜓𝜒)))

Theoremts3or1 33931 A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → (((𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑𝜓𝜒)))

Theoremts3or2 33932 A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → (¬ (𝜑𝜓) ∨ (𝜑𝜓𝜒)))

Theoremts3or3 33933 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 33934 Equality deduction for indexed union. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theoremabeq12 33935 Equality deduction for class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
(∀𝑥(𝜑𝜓) → {𝑥𝜑} = {𝑥𝜓})

Theoremrabeq12f 33936 Equality deduction for restricted class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
𝑥𝐴    &   𝑥𝐵       ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) → {𝑥𝐴𝜑} = {𝑥𝐵𝜓})

Theoremcsbeq12 33937 Equality deduction for substitution in class. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → 𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐷)

Theoremnfbii2 33938 Equality deduction for not-freeness. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
(∀𝑥(𝜑𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))

Theoremsbeqi 33939 Equality deduction for substitution. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
((𝑥 = 𝑦 ∧ ∀𝑧(𝜑𝜓)) → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜓))

Theoremralbi12f 33940 Equality deduction for restricted universal quantification. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
𝑥𝐴    &   𝑥𝐵       ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))

Theoremoprabbi 33941 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})

Theoremmpt2bi123f 33942* Equality deduction for maps-to notations with two arguments. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
𝑥𝐴    &   𝑥𝐵    &   𝑦𝐴    &   𝑦𝐵    &   𝑦𝐶    &   𝑦𝐷    &   𝑥𝐶    &   𝑥𝐷       (((𝐴 = 𝐵𝐶 = 𝐷) ∧ ∀𝑥𝐴𝑦𝐶 𝐸 = 𝐹) → (𝑥𝐴, 𝑦𝐶𝐸) = (𝑥𝐵, 𝑦𝐷𝐹))

Theoremiuneq12f 33943 Equality deduction for indexed unions. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
𝑥𝐴    &   𝑥𝐵       ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Theoremiineq12f 33944 Equality deduction for indexed intersections. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
𝑥𝐴    &   𝑥𝐵       ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Theoremopabbi 33945 Equality deduction for class abstraction of ordered pairs. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
(∀𝑥𝑦(𝜑𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})

Theoremmptbi12f 33946 Equality deduction for maps-to notations. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
𝑥𝐴    &   𝑥𝐵       ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐷 = 𝐸) → (𝑥𝐴𝐷) = (𝑥𝐵𝐸))

20.20.4  Miscellanea

Work in progress or things that do not belong anywhere else.

Theoremscottexf 33947* A version of scottex 8733 with non-free variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
𝑦𝐴    &   𝑥𝐴       {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V

Theoremscott0f 33948* A version of scott0 8734 with non-free variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
𝑦𝐴    &   𝑥𝐴       (𝐴 = ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)

Theoremscottn0f 33949* A version of scott0f 33948 with inequalities instead of equalities. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
𝑦𝐴    &   𝑥𝐴       (𝐴 ≠ ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅)

Theoremac6s3f 33950* Generalization of the Axiom of Choice to classes, with bound-variable hypothesis. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
𝑦𝜓    &   𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝜑 → ∃𝑓𝑥𝐴 𝜓)

Theoremac6s6 33951* Generalization of the Axiom of Choice to classes, moving the existence condition in the consequent. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
𝑦𝜓    &   𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       𝑓𝑥𝐴 (∃𝑦𝜑𝜓)

Theoremac6s6f 33952* Generalization of the Axiom of Choice to classes, moving the existence condition in the consequent. (Contributed by Giovanni Mascellani, 20-Aug-2018.)
𝐴 ∈ V    &   𝑦𝜓    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))    &   𝑥𝐴       𝑓𝑥𝐴 (∃𝑦𝜑𝜓)

20.21  Mathbox for Rodolfo Medina

20.21.1  Partitions

Theoremprtlem60 33953 Lemma for prter3 33986. (Contributed by Rodolfo Medina, 9-Oct-2010.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜓 → (𝜃𝜏))       (𝜑 → (𝜓 → (𝜒𝜏)))

Theorembicomdd 33954 Commute two sides of a biconditional in a deduction. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓 → (𝜃𝜒)))

Theoremjca2 33955 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.)
(𝜑 → (𝜓𝜒))    &   (𝜓𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremjca2r 33956 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 17-Oct-2010.)
(𝜑 → (𝜓𝜒))    &   (𝜓𝜃)       (𝜑 → (𝜓 → (𝜃𝜒)))

Theoremjca3 33957 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 14-Oct-2010.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜏)       (𝜑 → (𝜓 → (𝜃 → (𝜒𝜏))))

Theoremprtlem70 33958 Lemma for prter3 33986: a rearrangement of conjuncts. (Contributed by Rodolfo Medina, 20-Oct-2010.)
((((𝜓𝜂) ∧ ((𝜑𝜃) ∧ (𝜒𝜏))) ∧ 𝜑) ↔ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ (𝜃𝜏)))) ∧ 𝜂))

Theoremibdr 33959 Reverse of ibd 258. (Contributed by Rodolfo Medina, 30-Sep-2010.)
(𝜑 → (𝜒 → (𝜓𝜒)))       (𝜑 → (𝜒𝜓))

Theorempm5.31r 33960 Variant of pm5.31 611. (Contributed by Rodolfo Medina, 15-Oct-2010.)
((𝜒 ∧ (𝜑𝜓)) → (𝜑 → (𝜒𝜓)))

Theorem2r19.29 33961 Double the quantifiers of theorem r19.29. (Contributed by Rodolfo Medina, 25-Sep-2010.)
((∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → ∃𝑥𝐴𝑦𝐵 (𝜑𝜓))

Theoremprtlem100 33962 Lemma for prter3 33986. (Contributed by Rodolfo Medina, 19-Oct-2010.)
(∃𝑥𝐴 (𝐵𝑥𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵𝑥𝜑))

Theoremprtlem5 33963* Lemma for prter1 33983, prter2 33985, prter3 33986 and prtex 33984. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥))

Theoremprtlem80 33964 Lemma for prter2 33985. (Contributed by Rodolfo Medina, 17-Oct-2010.)
(𝐴𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))

Theoremn0el 33965* Negated membership of the empty set in another class. (Contributed by Rodolfo Medina, 25-Sep-2010.)
(¬ ∅ ∈ 𝐴 ↔ ∀𝑥𝐴𝑢 𝑢𝑥)

Theorembrabsb2 33966* A closed form of brabsb 4976. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑))

Theoremeqbrrdv2 33967* Other version of eqbrrdiv 5208. (Contributed by Rodolfo Medina, 30-Sep-2010.)
(((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦𝑥𝐵𝑦))       (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)

Theoremprtlem9 33968* Lemma for prter3 33986. (Contributed by Rodolfo Medina, 25-Sep-2010.)
(𝐴𝐵 → ∃𝑥𝐵 [𝑥] = [𝐴] )

Theoremprtlem10 33969* Lemma for prter3 33986. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
( Er 𝐴 → (𝑧𝐴 → (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧 ∈ [𝑣] 𝑤 ∈ [𝑣] ))))

Theoremprtlem11 33970 Lemma for prter2 33985. (Contributed by Rodolfo Medina, 12-Oct-2010.)
(𝐵𝐷 → (𝐶𝐴 → (𝐵 = [𝐶] 𝐵 ∈ (𝐴 / ))))

Theoremprtlem12 33971* Lemma for prtex 33984 and prter3 33986. (Contributed by Rodolfo Medina, 13-Oct-2010.)
( = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)} → Rel )

Theoremprtlem13 33972* Lemma for prter1 33983, prter2 33985, prter3 33986 and prtex 33984. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))

Theoremprtlem16 33973* Lemma for prtex 33984, prter2 33985 and prter3 33986. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       dom = 𝐴

Theoremprtlem400 33974* Lemma for prter2 33985 and also a property of partitions . (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}        ¬ ∅ ∈ ( 𝐴 / )

Syntaxwprt 33975 Extend the definition of a wff to include the partition predicate.
wff Prt 𝐴

Definitiondf-prt 33976* Define the partition predicate. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(Prt 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))

Theoremerprt 33977 The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.)
( Er 𝑋 → Prt (𝐴 / ))

Theoremprtlem14 33978* Lemma for prter1 33983, prter2 33985 and prtex 33984. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑤𝑥𝑤𝑦) → 𝑥 = 𝑦)))

Theoremprtlem15 33979* Lemma for prter1 33983 and prtex 33984. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(Prt 𝐴 → (∃𝑥𝐴𝑦𝐴 ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → ∃𝑧𝐴 (𝑢𝑧𝑣𝑧)))

Theoremprtlem17 33980* Lemma for prter2 33985. (Contributed by Rodolfo Medina, 15-Oct-2010.)
(Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) → 𝑤𝑥)))

Theoremprtlem18 33981* Lemma for prter2 33985. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (𝑤𝑣𝑧 𝑤)))

Theoremprtlem19 33982* Lemma for prter2 33985. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → 𝑣 = [𝑧] ))

Theoremprter1 33983* Every partition generates an equivalence relation. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 Er 𝐴)

Theoremprtex 33984* The equivalence relation generated by a partition is a set if and only if the partition itself is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ( ∈ V ↔ 𝐴 ∈ V))

Theoremprter2 33985* The quotient set of the equivalence relation generated by a partition equals the partition itself. (Contributed by Rodolfo Medina, 17-Oct-2010.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ( 𝐴 / ) = (𝐴 ∖ {∅}))

Theoremprter3 33986* For every partition there exists a unique equivalence relation whose quotient set equals the partition. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       ((𝑆 Er 𝐴 ∧ ( 𝐴 / 𝑆) = (𝐴 ∖ {∅})) → = 𝑆)

20.22  Mathbox for Norm Megill

Note: A label suffixed with "N" (after the "Atoms..." section below), such as lshpnel2N 34091, means that the definition or theorem is not used for the derivation of hlathil 37072. This is a temporary renaming to assist cleaning up the theorems needed by hlathil 37072.

20.22.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16

These older axiom schemes are obsolete and should not be used outside of this section. They are proved above as theorems axc4 , sp 2051, axc7 2130, axc10 2250, axc11 2312, axc11n 2305, axc15 2301, axc9 2300, axc14 2370, and axc16 2133.

Axiomax-c5 33987 Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all 𝑥, it is true for any specific 𝑥 (that would typically occur as a free variable in the wff substituted for 𝜑). (A free variable is one that does not occur in the scope of a quantifier: 𝑥 and 𝑦 are both free in 𝑥 = 𝑦, but only 𝑥 is free in 𝑦𝑥 = 𝑦.) Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77).

Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1720. Conditional forms of the converse are given by ax-13 2244, ax-c14 33995, ax-c16 33996, and ax-5 1837.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from 𝑥 for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 2351.

An interesting alternate axiomatization uses axc5c711 34022 and ax-c4 33988 in place of ax-c5 33987, ax-4 1735, ax-10 2017, and ax-11 2032.

This axiom is obsolete and should no longer be used. It is proved above as theorem sp 2051. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)

(∀𝑥𝜑𝜑)

Axiomax-c4 33988 Axiom of Quantified Implication. This axiom moves a quantifier from outside to inside an implication, quantifying 𝜓. Notice that 𝑥 must not be a free variable in the antecedent of the quantified implication, and we express this by binding 𝜑 to "protect" the axiom from a 𝜑 containing a free 𝑥. Axiom scheme C4' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Lemma 5 of [Monk2] p. 108 and Axiom 5 of [Mendelson] p. 69.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc4 2128. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)

(∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Axiomax-c7 33989 Axiom of Quantified Negation. This axiom is used to manipulate negated quantifiers. Equivalent to axiom scheme C7' in [Megill] p. 448 (p. 16 of the preprint). An alternate axiomatization could use axc5c711 34022 in place of ax-c5 33987, ax-c7 33989, and ax-11 2032.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc7 2130. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Axiomax-c10 33990 A variant of ax6 2249. Axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint).

This axiom is obsolete and should no longer be used. It is proved above as theorem axc10 2250. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Axiomax-c11 33991 Axiom ax-c11 33991 was the original version of ax-c11n 33992 ("n" for "new"), before it was discovered (in May 2008) that the shorter ax-c11n 33992 could replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of the preprint).

This axiom is obsolete and should no longer be used. It is proved above as theorem axc11 2312. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Axiomax-c11n 33992 Axiom of Quantifier Substitution. One of the equality and substitution axioms of predicate calculus with equality. Appears as Lemma L12 in [Megill] p. 445 (p. 12 of the preprint).

The original version of this axiom was ax-c11 33991 and was replaced with this shorter ax-c11n 33992 ("n" for "new") in May 2008. The old axiom is proved from this one as theorem axc11 2312. Conversely, this axiom is proved from ax-c11 33991 as theorem axc11nfromc11 34030.

This axiom was proved redundant in July 2015. See theorem axc11n 2305.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc11n 2305. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Axiomax-c15 33993 Axiom ax-c15 33993 was the original version of ax-12 2045, before it was discovered (in Jan. 2007) that the shorter ax-12 2045 could replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of the preprint). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases. To understand this theorem more easily, think of "¬ ∀𝑥𝑥 = 𝑦..." as informally meaning "if 𝑥 and 𝑦 are distinct variables then..." The antecedent becomes false if the same variable is substituted for 𝑥 and 𝑦, ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form ¬ ∀𝑥𝑥 = 𝑦 a "distinctor."

Interestingly, if the wff expression substituted for 𝜑 contains no wff variables, the resulting statement can be proved without invoking this axiom. This means that even though this axiom is metalogically independent from the others, it is not logically independent. Specifically, we can prove any wff-variable-free instance of axiom ax-c15 33993 (from which the ax-12 2045 instance follows by theorem ax12 2302.) The proof is by induction on formula length, using ax12eq 34045 and ax12el 34046 for the basis steps and ax12indn 34047, ax12indi 34048, and ax12inda 34052 for the induction steps. (This paragraph is true provided we use ax-c11 33991 in place of ax-c11n 33992.)

This axiom is obsolete and should no longer be used. It is proved above as theorem axc15 2301, which should be used instead. (Contributed by NM, 14-May-1993.) (New usage is discouraged.)

(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Axiomax-c9 33994 Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever 𝑧 is distinct from 𝑥 and 𝑦, and 𝑥 = 𝑦 is true, then 𝑥 = 𝑦 quantified with 𝑧 is also true. In other words, 𝑧 is irrelevant to the truth of 𝑥 = 𝑦. Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc9 2300. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Axiomax-c14 33995 Axiom of Quantifier Introduction. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. Axiom scheme C14' in [Megill] p. 448 (p. 16 of the preprint). It is redundant if we include ax-5 1837; see theorem axc14 2370. Alternately, ax-5 1837 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-5 1837. We retain ax-c14 33995 here to provide completeness for systems with the simpler metalogic that results from omitting ax-5 1837, which might be easier to study for some theoretical purposes.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc14 2370. (Contributed by NM, 24-Jun-1993.) (New usage is discouraged.)

(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))

Axiomax-c16 33996* Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-5 1837 to be a metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases. It is a somewhat bizarre axiom since the antecedent is always false in set theory (see dtru 4848), but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-5 1837; see theorem axc16 2133. Alternately, ax-5 1837 becomes logically redundant in the presence of this axiom, but without ax-5 1837 we lose the more powerful metalogic that results from being able to express the concept of a setvar variable not occurring in a wff (as opposed to just two setvar variables being distinct). We retain ax-c16 33996 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-5 1837, which might be easier to study for some theoretical purposes.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc16 2133. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

20.22.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old

Theorems ax12fromc15 34009 and ax13fromc9 34010 require some intermediate theorems that are included in this section.

Theoremaxc5 33997 This theorem repeats sp 2051 under the name axc5 33997, so that the metamath program's "verify markup" command will check that it matches axiom scheme ax-c5 33987. It is preferred that references to this theorem use the name sp 2051. (Contributed by NM, 18-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥𝜑𝜑)

Theoremax4fromc4 33998 Rederivation of axiom ax-4 1735 from ax-c4 33988, ax-c5 33987, ax-gen 1720 and minimal implicational calculus { ax-mp 5, ax-1 6, ax-2 7 }. See axc4 2128 for the derivation of ax-c4 33988 from ax-4 1735. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Theoremax10fromc7 33999 Rederivation of axiom ax-10 2017 from ax-c7 33989, ax-c4 33988, ax-c5 33987, ax-gen 1720 and propositional calculus. See axc7 2130 for the derivation of ax-c7 33989 from ax-10 2017. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremax6fromc10 34000 Rederivation of axiom ax-6 1886 from ax-c7 33989, ax-c10 33990, ax-gen 1720 and propositional calculus. See axc10 2250 for the derivation of ax-c10 33990 from ax-6 1886. Lemma L18 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ∀𝑥 ¬ 𝑥 = 𝑦

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