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Theorem List for Metamath Proof Explorer - 33001-33100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtsxo4 33001 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑𝜓) ∨ (𝜑𝜓)))

Theoremtsan1 33002 A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))

Theoremtsan2 33003 A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜑 ∨ ¬ (𝜑𝜓)))

Theoremtsan3 33004 A Tseitin axiom for logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜓 ∨ ¬ (𝜑𝜓)))

Theoremtsna1 33005 A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsna2 33006 A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜑 ∨ (𝜑𝜓)))

Theoremtsna3 33007 A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(𝜃 → (𝜓 ∨ (𝜑𝜓)))

Theoremtsor1 33008 A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Theoremtsor2 33009 A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → (¬ 𝜑 ∨ (𝜑𝜓)))

Theoremtsor3 33010 A Tseitin axiom for logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → (¬ 𝜓 ∨ (𝜑𝜓)))

Theoremts3an1 33011 A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → ((¬ (𝜑𝜓) ∨ ¬ 𝜒) ∨ (𝜑𝜓𝜒)))

Theoremts3an2 33012 A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → ((𝜑𝜓) ∨ ¬ (𝜑𝜓𝜒)))

Theoremts3an3 33013 A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → (𝜒 ∨ ¬ (𝜑𝜓𝜒)))

Theoremts3or1 33014 A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → (((𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑𝜓𝜒)))

Theoremts3or2 33015 A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
(𝜃 → (¬ (𝜑𝜓) ∨ (𝜑𝜓𝜒)))

Theoremts3or3 33016 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 33017 Equality deduction for indexed union. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theoremabeq12 33018 Equality deduction for class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
(∀𝑥(𝜑𝜓) → {𝑥𝜑} = {𝑥𝜓})

Theoremrabeq12f 33019 Equality deduction for restricted class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
𝑥𝐴    &   𝑥𝐵       ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) → {𝑥𝐴𝜑} = {𝑥𝐵𝜓})

Theoremcsbeq12 33020 Equality deduction for substitution in class. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → 𝐴 / 𝑥𝐶 = 𝐵 / 𝑥𝐷)

Theoremnfbii2 33021 Equality deduction for not-freeness. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
(∀𝑥(𝜑𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))

Theoremsbeqi 33022 Equality deduction for substitution. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
((𝑥 = 𝑦 ∧ ∀𝑧(𝜑𝜓)) → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜓))

Theoremralbi12f 33023 Equality deduction for restricted universal quantification. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
𝑥𝐴    &   𝑥𝐵       ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))

Theoremoprabbi 33024 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 33025* 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 33026 Equality deduction for indexed unions. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
𝑥𝐴    &   𝑥𝐵       ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Theoremiineq12f 33027 Equality deduction for indexed intersections. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
𝑥𝐴    &   𝑥𝐵       ((𝐴 = 𝐵 ∧ ∀𝑥𝐴 𝐶 = 𝐷) → 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Theoremopabbi 33028 Equality deduction for class abstraction of ordered pairs. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
(∀𝑥𝑦(𝜑𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓})

Theoremmptbi12f 33029 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 33030* A version of scottex 8511 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 33031* A version of scott0 8512 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 33032* A version of scott0f 33031 with inequalities instead of equalities. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
𝑦𝐴    &   𝑥𝐴       (𝐴 ≠ ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ≠ ∅)

Theoremac6s3f 33033* 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 33034* 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 33035* 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 33036 Lemma for prter3 33069. (Contributed by Rodolfo Medina, 9-Oct-2010.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜓 → (𝜃𝜏))       (𝜑 → (𝜓 → (𝜒𝜏)))

Theorembicomdd 33037 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 33038 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.)
(𝜑 → (𝜓𝜒))    &   (𝜓𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremjca2r 33039 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 17-Oct-2010.)
(𝜑 → (𝜓𝜒))    &   (𝜓𝜃)       (𝜑 → (𝜓 → (𝜃𝜒)))

Theoremjca3 33040 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 14-Oct-2010.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜏)       (𝜑 → (𝜓 → (𝜃 → (𝜒𝜏))))

Theoremprtlem70 33041 Lemma for prter3 33069: a rearrangement of conjuncts. (Contributed by Rodolfo Medina, 20-Oct-2010.)
((((𝜓𝜂) ∧ ((𝜑𝜃) ∧ (𝜒𝜏))) ∧ 𝜑) ↔ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ (𝜃𝜏)))) ∧ 𝜂))

Theoremibdr 33042 Reverse of ibd 256. (Contributed by Rodolfo Medina, 30-Sep-2010.)
(𝜑 → (𝜒 → (𝜓𝜒)))       (𝜑 → (𝜒𝜓))

Theorempm5.31r 33043 Variant of pm5.31 609. (Contributed by Rodolfo Medina, 15-Oct-2010.)
((𝜒 ∧ (𝜑𝜓)) → (𝜑 → (𝜒𝜓)))

Theorem2r19.29 33044 Double the quantifiers of theorem r19.29. (Contributed by Rodolfo Medina, 25-Sep-2010.)
((∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → ∃𝑥𝐴𝑦𝐵 (𝜑𝜓))

Theoremprtlem100 33045 Lemma for prter3 33069. (Contributed by Rodolfo Medina, 19-Oct-2010.)
(∃𝑥𝐴 (𝐵𝑥𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵𝑥𝜑))

Theoremprtlem5 33046* Lemma for prter1 33066, prter2 33068, prter3 33069 and prtex 33067. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥𝐴 (𝑢𝑥𝑣𝑥) ↔ ∃𝑥𝐴 (𝑟𝑥𝑠𝑥))

Theoremprtlem80 33047 Lemma for prter2 33068. (Contributed by Rodolfo Medina, 17-Oct-2010.)
(𝐴𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))

Theoremn0el 33048* Negated membership of the empty set in another class. (Contributed by Rodolfo Medina, 25-Sep-2010.)
(¬ ∅ ∈ 𝐴 ↔ ∀𝑥𝐴𝑢 𝑢𝑥)

Theorembrabsb2 33049* A closed form of brabsb 4805. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑))

Theoremeqbrrdv2 33050* Other version of eqbrrdiv 5034. (Contributed by Rodolfo Medina, 30-Sep-2010.)
(((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦𝑥𝐵𝑦))       (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)

Theoremprtlem9 33051* Lemma for prter3 33069. (Contributed by Rodolfo Medina, 25-Sep-2010.)
(𝐴𝐵 → ∃𝑥𝐵 [𝑥] = [𝐴] )

Theoremprtlem10 33052* Lemma for prter3 33069. (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 33053 Lemma for prter2 33068. (Contributed by Rodolfo Medina, 12-Oct-2010.)
(𝐵𝐷 → (𝐶𝐴 → (𝐵 = [𝐶] 𝐵 ∈ (𝐴 / ))))

Theoremprtlem12 33054* Lemma for prtex 33067 and prter3 33069. (Contributed by Rodolfo Medina, 13-Oct-2010.)
( = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)} → Rel )

Theoremprtlem13 33055* Lemma for prter1 33066, prter2 33068, prter3 33069 and prtex 33067. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (𝑧 𝑤 ↔ ∃𝑣𝐴 (𝑧𝑣𝑤𝑣))

Theoremprtlem16 33056* Lemma for prtex 33067, prter2 33068 and prter3 33069. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       dom = 𝐴

Theoremprtlem400 33057* Lemma for prter2 33068 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 33058 Extend the definition of a wff to include the partition predicate.
wff Prt 𝐴

Definitiondf-prt 33059* Define the partition predicate. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(Prt 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))

Theoremerprt 33060 The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.)
( Er 𝑋 → Prt (𝐴 / ))

Theoremprtlem14 33061* Lemma for prter1 33066, prter2 33068 and prtex 33067. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(Prt 𝐴 → ((𝑥𝐴𝑦𝐴) → ((𝑤𝑥𝑤𝑦) → 𝑥 = 𝑦)))

Theoremprtlem15 33062* Lemma for prter1 33066 and prtex 33067. (Contributed by Rodolfo Medina, 13-Oct-2010.)
(Prt 𝐴 → (∃𝑥𝐴𝑦𝐴 ((𝑢𝑥𝑤𝑥) ∧ (𝑤𝑦𝑣𝑦)) → ∃𝑧𝐴 (𝑢𝑧𝑣𝑧)))

Theoremprtlem17 33063* Lemma for prter2 33068. (Contributed by Rodolfo Medina, 15-Oct-2010.)
(Prt 𝐴 → ((𝑥𝐴𝑧𝑥) → (∃𝑦𝐴 (𝑧𝑦𝑤𝑦) → 𝑤𝑥)))

Theoremprtlem18 33064* Lemma for prter2 33068. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → (𝑤𝑣𝑧 𝑤)))

Theoremprtlem19 33065* Lemma for prter2 33068. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
= {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}       (Prt 𝐴 → ((𝑣𝐴𝑧𝑣) → 𝑣 = [𝑧] ))

Theoremprter1 33066* 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 33067* 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 33068* 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 33069* 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 33174, means that the definition or theorem is not used for the derivation of hlathil 36155. This is a temporary renaming to assist cleaning up the theorems needed by hlathil 36155.

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 1990, axc7 1992, axc10 2143, axc11 2206, axc11n 2199, axc15 2195, axc9 2194, axc14 2264, and axc16 2072.

Axiomax-c5 33070 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 1700. Conditional forms of the converse are given by ax-13 2137, ax-c14 33078, ax-c16 33079, and ax-5 1793.

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 2245.

An interesting alternate axiomatization uses axc5c711 33105 and ax-c4 33071 in place of ax-c5 33070, ax-4 1713, ax-10 1966, and ax-11 1971.

This axiom is obsolete and should no longer be used. It is proved above as theorem sp 1990. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)

(∀𝑥𝜑𝜑)

Axiomax-c4 33071 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 1991. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)

(∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Axiomax-c7 33072 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 33105 in place of ax-c5 33070, ax-c7 33072, and ax-11 1971.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc7 1992. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Axiomax-c10 33073 A variant of ax6 2142. 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 2143. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Axiomax-c11 33074 Axiom ax-c11 33074 was the original version of ax-c11n 33075 ("n" for "new"), before it was discovered (in May 2008) that the shorter ax-c11n 33075 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 2206. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Axiomax-c11n 33075 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 33074 and was replaced with this shorter ax-c11n 33075 ("n" for "new") in May 2008. The old axiom is proved from this one as theorem axc11 2206. Conversely, this axiom is proved from ax-c11 33074 as theorem axc11nfromc11 33113.

This axiom was proved redundant in July 2015. See theorem axc11n 2199.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc11n 2199. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Axiomax-c15 33076 Axiom ax-c15 33076 was the original version of ax-12 1983, before it was discovered (in Jan. 2007) that the shorter ax-12 1983 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 33076 (from which the ax-12 1983 instance follows by theorem ax12 2196.) The proof is by induction on formula length, using ax12eq 33128 and ax12el 33129 for the basis steps and ax12indn 33130, ax12indi 33131, and ax12inda 33135 for the induction steps. (This paragraph is true provided we use ax-c11 33074 in place of ax-c11n 33075.)

This axiom is obsolete and should no longer be used. It is proved above as theorem axc15 2195, 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 33077 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 2194. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

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

Axiomax-c14 33078 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 1793; see theorem axc14 2264. Alternately, ax-5 1793 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-5 1793. We retain ax-c14 33078 here to provide completeness for systems with the simpler metalogic that results from omitting ax-5 1793, 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 2264. (Contributed by NM, 24-Jun-1993.) (New usage is discouraged.)

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

Axiomax-c16 33079* Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-5 1793 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 4682), but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-5 1793; see theorem axc16 2072. Alternately, ax-5 1793 becomes logically redundant in the presence of this axiom, but without ax-5 1793 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 33079 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-5 1793, 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 2072. (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 33092 and ax13fromc9 33093 require some intermediate theorems that are included in this section.

Theoremaxc5 33080 This theorem repeats sp 1990 under the name axc5 33080, so that the metamath program's "verify markup" command will check that it matches axiom scheme ax-c5 33070. It is preferred that references to this theorem use the name sp 1990. (Contributed by NM, 18-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥𝜑𝜑)

Theoremax4fromc4 33081 Rederivation of axiom ax-4 1713 from ax-c4 33071, ax-c5 33070, ax-gen 1700 and minimal implicational calculus { ax-mp 5, ax-1 6, ax-2 7 }. See axc4 1991 for the derivation of ax-c4 33071 from ax-4 1713. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Theoremax10fromc7 33082 Rederivation of axiom ax-10 1966 from ax-c7 33072, ax-c4 33071, ax-c5 33070, ax-gen 1700 and propositional calculus. See axc7 1992 for the derivation of ax-c7 33072 from ax-10 1966. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremax6fromc10 33083 Rederivation of axiom ax-6 1838 from ax-c7 33072, ax-c10 33073, ax-gen 1700 and propositional calculus. See axc10 2143 for the derivation of ax-c10 33073 from ax-6 1838. 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

Theoremhba1-o 33084 The setvar 𝑥 is not free in 𝑥𝜑. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 24-Jan-1993.) (New usage is discouraged.)
(∀𝑥𝜑 → ∀𝑥𝑥𝜑)

Theoremaxc4i-o 33085 Inference version of ax-c4 33071. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.)
(∀𝑥𝜑𝜓)       (∀𝑥𝜑 → ∀𝑥𝜓)

Theoremequid1 33086 Proof of equid 1889 from our older axioms. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms (although the proof, as you can see below, is not as obvious as you might think). This proof uses only axioms without distinct variable conditions and requires no dummy variables. A simpler proof, similar to Tarski's, is possible if we make use of ax-5 1793; see the proof of equid 1889. See equid1ALT 33112 for an alternate proof. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 = 𝑥

Theoremequcomi1 33087 Proof of equcomi 1894 from equid1 33086, avoiding use of ax-5 1793 (the only use of ax-5 1793 is via ax7 1893, so using ax-7 1885 instead would remove dependency on ax-5 1793). (Contributed by BJ, 8-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremaecom-o 33088 Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version of aecom 2203 using ax-c11 33074. Unlike axc11nfromc11 33113, this version does not require ax-5 1793 (see comment of equcomi1 33087). (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaecoms-o 33089 A commutation rule for identical variable specifiers. Version of aecoms 2204 using ax-c11 33074. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦𝜑)       (∀𝑦 𝑦 = 𝑥𝜑)

Theoremhbae-o 33090 All variables are effectively bound in an identical variable specifier. Version of hbae 2207 using ax-c11 33074. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)

Theoremdral1-o 33091 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral1 2217 using ax-c11 33074. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Theoremax12fromc15 33092 Rederivation of axiom ax-12 1983 from ax-c15 33076, ax-c11 33074 (used through dral1-o 33091), and other older axioms. See theorem axc15 2195 for the derivation of ax-c15 33076 from ax-12 1983.

An open problem is whether we can prove this using ax-c11n 33075 instead of ax-c11 33074.

This proof uses newer axioms ax-4 1713 and ax-6 1838, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 33071 and ax-c10 33073. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax13fromc9 33093 Derive ax-13 2137 from ax-c9 33077 and other older axioms.

This proof uses newer axioms ax-4 1713 and ax-6 1838, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 33071 and ax-c10 33073. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

20.22.3  Legacy theorems using obsolete axioms

These theorems were mostly intended to study properties of the older axiom schemes and are not useful outside of this section. They should not be used outside of this section. They may be deleted when they are deemed to no longer be of interest.

Theoremax5ALT 33094* Axiom to quantify a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113.

(This theorem simply repeats ax-5 1793 so that we can include the following note, which applies only to the obsolete axiomatization.)

This axiom is logically redundant in the (logically complete) predicate calculus axiom system consisting of ax-gen 1700, ax-c4 33071, ax-c5 33070, ax-11 1971, ax-c7 33072, ax-7 1885, ax-c9 33077, ax-c10 33073, ax-c11 33074, ax-8 1940, ax-9 1947, ax-c14 33078, ax-c15 33076, and ax-c16 33079: in that system, we can derive any instance of ax-5 1793 not containing wff variables by induction on formula length, using ax5eq 33119 and ax5el 33124 for the basis together with hbn 2025, hbal 1973, and hbim 2053. However, if we omit this axiom, our development would be quite inconvenient since we could work only with specific instances of wffs containing no wff variables - this axiom introduces the concept of a setvar variable not occurring in a wff (as opposed to just two setvar variables being distinct). (Contributed by NM, 19-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

(𝜑 → ∀𝑥𝜑)

Theoremsps-o 33095 Generalization of antecedent. (Contributed by NM, 5-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theoremhbequid 33096 Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (The proof does not use ax-c10 33073.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)

Theoremnfequid-o 33097 Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (The proof uses only ax-4 1713, ax-7 1885, ax-c9 33077, and ax-gen 1700. This shows that this can be proved without ax6 2142, even though the theorem equid 1889 cannot be. A shorter proof using ax6 2142 is obtainable from equid 1889 and hbth 1706.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax6v 1839, which is used for the derivation of axc9 2194, unless we consider ax-c9 33077 the starting axiom rather than ax-13 2137. (Contributed by NM, 13-Jan-2011.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦 𝑥 = 𝑥

Theoremaxc5c7 33098 Proof of a single axiom that can replace ax-c5 33070 and ax-c7 33072. See axc5c7toc5 33099 and axc5c7toc7 33100 for the rederivation of those axioms. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑) → 𝜑)

Theoremaxc5c7toc5 33099 Rederivation of ax-c5 33070 from axc5c7 33098. Only propositional calculus is used for the rederivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑𝜑)

Theoremaxc5c7toc7 33100 Rederivation of ax-c7 33072 from axc5c7 33098. Only propositional calculus is used for the rederivation. (Contributed by Scott Fenton, 12-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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