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Theorem List for Metamath Proof Explorer - 2301-2400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhbim1 2301 A closed form of hbim 2303. (Contributed by NM, 2-Jun-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theoremhbimd 2302 Deduction form of bound-variable hypothesis builder hbim 2303. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))       (𝜑 → ((𝜓𝜒) → ∀𝑥(𝜓𝜒)))

Theoremhbim 2303 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by NM, 24-Jan-1993.) (Proof shortened by Mel L. O'Cat, 3-Mar-2008.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theoremhban 2304 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theoremhb3an 2305 If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑𝜓𝜒). (Contributed by NM, 14-Sep-2003.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝜒 → ∀𝑥𝜒)       ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))

Theoremsbi2 2306 Introduction of implication into substitution. (Contributed by NM, 14-May-1993.)
(([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))

Theoremsbim 2307 Implication inside and outside of a substitution are equivalent. (Contributed by NM, 14-May-1993.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

TheoremsbanOLD 2308 Obsolete version of sban 2085 as of 13-Aug-2023. Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-May-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))

Theoremsbrim 2309 Substitution in an implication with a variable not free in the antecedent affects only the consequent. See sbrimv 2310 for a version with disjoint variables not requiring ax-10 2142. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑥𝜑       ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

Theoremsbrimv 2310* Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim 2309 not depending on ax-10 2142, but with disjoint variables. (Contributed by Wolf Lammen, 28-Jan-2024.)
𝑥𝜑       ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

Theoremsblim 2311 Substitution in an implication with a variable not free in the consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑥𝜓       ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓))

Theoremsbor 2312 Disjunction inside and outside of a substitution are equivalent. (Contributed by NM, 29-Sep-2002.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))

Theoremsbbi 2313 Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 14-May-1993.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))

Theoremsblbis 2314 Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.)
([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥](𝜒𝜑) ↔ ([𝑦 / 𝑥]𝜒𝜓))

Theoremsbrbis 2315 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥](𝜑𝜒) ↔ (𝜓 ↔ [𝑦 / 𝑥]𝜒))

Theoremsbrbif 2316 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑥𝜒    &   ([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥](𝜑𝜒) ↔ (𝜓𝜒))

Theoremsbi1vOLD 2317* Obsolete version of sbi1 2076 as of 24-Jul-2023. Move implication out of substitution. Version of sbi1 2076 with a disjoint variable condition, not requiring ax-13 2379. (Contributed by Wolf Lammen, 18-Jan-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

TheoremsbimvOLD 2318* Obsolete version of sbim 2307 as of 24-Jul-2023. Substitution distributes over implication. Version of sbim 2307 with a disjoint variable condition, not requiring ax-13 2379. (Contributed by Wolf Lammen, 18-Jan-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

TheoremsbanvOLD 2319* Obsolete version of sban 2085 as of 24-Jul-2023. Substitution distributes over conjunction. Version of sban 2085 with a disjoint variable condition, not requiring ax-13 2379. (Contributed by Wolf Lammen, 18-Jan-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))

TheoremsbbivOLD 2320* Obsolete version of sbbi 2313 as of 24-Jul-2023. Substitution distributes over a biconditional. Version of sbbi 2313 with a disjoint variable condition, not requiring ax-11 2158 nor ax-13 2379. (Contributed by Wolf Lammen, 18-Jan-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))

TheoremsblbisvOLD 2321* Obsolete version of sblbis 2314 as of 24-Jul-2023. Introduce left biconditional inside of a substitution. Version of sblbis 2314 with a disjoint variable condition, not requiring ax-13 2379. (Contributed by Wolf Lammen, 18-Jan-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥](𝜒𝜑) ↔ ([𝑦 / 𝑥]𝜒𝜓))

Theoremsbiev 2322* Conversion of implicit substitution to explicit substitution. Version of sbie 2521 with a disjoint variable condition, not requiring ax-13 2379. See sbievw 2100 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Wolf Lammen, 18-Jan-2023.) Remove dependence on ax-10 2142 and shorten proof. (Revised by BJ, 18-Jul-2023.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       ([𝑦 / 𝑥]𝜑𝜓)

Theoremsbiedw 2323* Conversion of implicit substitution to explicit substitution (deduction version of sbiev 2322). Version of sbied 2522 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-10 2142. (Revised by Wolf Lammen, 28-Jan-2024.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))

TheoremsbiedwOLD 2324* Obsolete version of sbiedw 2323 as of 28-Jan-2024. (Contributed by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))

Theoremaxc7 2325 Show that the original axiom ax-c7 36200 can be derived from ax-10 2142 (hbn1 2143) , sp 2180 and propositional calculus. See ax10fromc7 36210 for the rederivation of ax-10 2142 from ax-c7 36200.

Normally, axc7 2325 should be used rather than ax-c7 36200, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.)

(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)

Theoremaxc7e 2326 Abbreviated version of axc7 2325 using the existential quantifier. Corresponds to the dual of Axiom (B) of modal logic. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Jul-2022.)
(∃𝑥𝑥𝜑𝜑)

Theoremmodal-b 2327 The analogue in our predicate calculus of the Brouwer axiom (B) of modal logic S5. (Contributed by NM, 5-Oct-2005.)
(𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)

Theorem19.9ht 2328 A closed version of 19.9h 2290. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))

Theoremaxc4 2329 Show that the original axiom ax-c4 36199 can be derived from ax-4 1811 (alim 1812), ax-10 2142 (hbn1 2143), sp 2180 and propositional calculus. See ax4fromc4 36209 for the rederivation of ax-4 1811 from ax-c4 36199.

Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.)

(∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Theoremaxc4i 2330 Inference version of axc4 2329. (Contributed by NM, 3-Jan-1993.)
(∀𝑥𝜑𝜓)       (∀𝑥𝜑 → ∀𝑥𝜓)

Theoremnfal 2331 If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       𝑥𝑦𝜑

Theoremnfex 2332 If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Reduce symbol count in nfex 2332, hbex 2333. (Revised by Wolf Lammen, 16-Oct-2021.)
𝑥𝜑       𝑥𝑦𝜑

Theoremhbex 2333 If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by NM, 12-Mar-1993.) Reduce symbol count in nfex 2332, hbex 2333. (Revised by Wolf Lammen, 16-Oct-2021.)
(𝜑 → ∀𝑥𝜑)       (∃𝑦𝜑 → ∀𝑥𝑦𝜑)

Theoremnfnf 2334 If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
𝑥𝜑       𝑥𝑦𝜑

Theorem19.12 2335 Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! But sometimes the converse does hold, as in 19.12vv 2357 and r19.12sn 4616. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
(∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremnfald 2336 Deduction form of nfal 2331. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 16-Oct-2021.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)

Theoremnfexd 2337 If 𝑥 is not free in 𝜓, it is not free in 𝑦𝜓. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)

Theoremnfsbv 2338* If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑧 is distinct from 𝑥 and 𝑦. Version of nfsb 2542 requiring more disjoint variables, but fewer axioms. (Contributed by Mario Carneiro, 11-Aug-2016.) (Revised by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on 𝑥, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.)
𝑧𝜑       𝑧[𝑦 / 𝑥]𝜑

TheoremnfsbvOLD 2339* Obsolete version of nfsbv 2338 as of 13-Aug-2023. (Contributed by Wolf Lammen, 7-Feb-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑧𝜑       𝑧[𝑦 / 𝑥]𝜑

TheoremhbsbwOLD 2340* Obsolete version of hbsbw 2173 as of 23-May-2024. (Contributed by NM, 12-Aug-1993.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑧𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)

Theoremsbco2v 2341* A composition law for substitution. Version of sbco2 2530 with disjoint variable conditions, not requiring ax-13 2379, but ax-11 2158. (Contributed by NM, 30-Jun-1994.) (Revised by Wolf Lammen, 29-Apr-2023.)
𝑧𝜑       ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Theoremaaan 2342 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
𝑦𝜑    &   𝑥𝜓       (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))

Theoremeeor 2343 Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))

Theoremcbv3v 2344* Rule used to change bound variables, using implicit substitution. Version of cbv3 2404 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremcbv1v 2345* Rule used to change bound variables, using implicit substitution. Version of cbv1 2411 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 16-Jun-2019.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Theoremcbv2w 2346* Rule used to change bound variables, using implicit substitution. Version of cbv2 2412 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 5-Aug-1993.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbvaldw 2347* Deduction used to change bound variables, using implicit substitution. Version of cbvald 2417 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 2-Jan-2002.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbvexdw 2348* Deduction used to change bound variables, using implicit substitution. Version of cbvexd 2418 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 2-Jan-2002.) (Revised by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Theoremcbv3hv 2349* Rule used to change bound variables, using implicit substitution. Version of cbv3h 2413 with a disjoint variable condition on 𝑥, 𝑦, which does not require ax-13 2379. Was used in a proof of axc11n 2437 (but of independent interest). (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Nov-2020.) (Proof shortened by BJ, 30-Nov-2020.)
(𝜑 → ∀𝑦𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremcbvalv1 2350* Rule used to change bound variables, using implicit substitution. Version of cbval 2405 with a disjoint variable condition, which does not require ax-13 2379. See cbvalvw 2043 for a version with two disjoint variable conditions, requiring fewer axioms, and cbvalv 2407 for another variant. (Contributed by NM, 13-May-1993.) (Revised by BJ, 31-May-2019.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremcbvexv1 2351* Rule used to change bound variables, using implicit substitution. Version of cbvex 2406 with a disjoint variable condition, which does not require ax-13 2379. See cbvexvw 2044 for a version with two disjoint variable conditions, requiring fewer axioms, and cbvexv 2408 for another variant. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 31-May-2019.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theoremcbval2v 2352* Rule used to change bound variables, using implicit substitution. Version of cbval2 2421 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 22-Dec-2003.) (Revised by BJ, 16-Jun-2019.) (Proof shortened by Gino Giotto, 10-Jan-2024.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theoremcbval2vOLD 2353* Obsolete version of cbval2v 2352 as of 14-Jan-2024. (Contributed by BJ, 16-Jan-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theoremcbvex2v 2354* Rule used to change bound variables, using implicit substitution. Version of cbvex2 2423 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 14-Sep-2003.) (Revised by BJ, 16-Jun-2019.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theoremdvelimhw 2355* Proof of dvelimh 2461 without using ax-13 2379 but with additional distinct variable conditions. (Contributed by NM, 1-Oct-2002.) (Revised by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 23-Dec-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑧𝜓)    &   (𝑧 = 𝑦 → (𝜑𝜓))    &   (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Theorempm11.53 2356* Theorem *11.53 in [WhiteheadRussell] p. 164. See pm11.53v 1945 for a version requiring fewer axioms. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))

Theorem19.12vv 2357* Special case of 19.12 2335 where its converse holds. See 19.12vvv 1995 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.)
(∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))

Theoremeean 2358 Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Theoremeeanv 2359* Distribute a pair of existential quantifiers over a conjunction. Combination of 19.41v 1950 and 19.42v 1954. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1956. (Contributed by NM, 26-Jul-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Theoremeeeanv 2360* Distribute three existential quantifiers over a conjunction. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) Reduce distinct variable restrictions. (Revised by Wolf Lammen, 20-Jan-2018.)
(∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒))

Theoremee4anv 2361* Distribute two pairs of existential quantifiers over a conjunction. For a version requiring fewer axioms but with additional disjoint variable conditions, see 4exdistrv 1957. (Contributed by NM, 31-Jul-1995.)
(∃𝑥𝑦𝑧𝑤(𝜑𝜓) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝜓))

Theoremsb8v 2362* Substitution of variable in universal quantifier. Version of sb8 2536 with a disjoint variable condition, not requiring ax-13 2379. (Contributed by NM, 16-May-1993.) (Revised by Wolf Lammen, 19-Jan-2023.)
𝑦𝜑       (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Theoremsb8ev 2363* Substitution of variable in existential quantifier. Version of sb8e 2537 with a disjoint variable condition, not requiring ax-13 2379. (Contributed by NM, 12-Aug-1993.) (Revised by Wolf Lammen, 19-Jan-2023.)
𝑦𝜑       (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Theorem2sb8ev 2364* An equivalent expression for double existence. Version of 2sb8e 2552 with more disjoint variable conditions, not requiring ax-13 2379. (Contributed by Wolf Lammen, 28-Jan-2023.)
𝑤𝜑    &   𝑧𝜑       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

Theoremsb6rfv 2365* Reversed substitution. Version of sb6rf 2480 requiring disjoint variables, but fewer axioms. (Contributed by NM, 1-Aug-1993.) (Revised by Wolf Lammen, 7-Feb-2023.)
𝑦𝜑       (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))

Theoremsbnf2 2366* Two ways of expressing "𝑥 is (effectively) not free in 𝜑". (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Sep-2018.) Avoid ax-13 2379. (Revised by Wolf Lammen, 30-Jan-2023.)
(Ⅎ𝑥𝜑 ↔ ∀𝑦𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))

Theoremexsb 2367* An equivalent expression for existence. One direction (exsbim 2008) needs fewer axioms. (Contributed by NM, 2-Feb-2005.) Avoid ax-13 2379. (Revised by Wolf Lammen, 16-Oct-2022.)
(∃𝑥𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝑦𝜑))

Theorem2exsb 2368* An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 30-Sep-2018.)
(∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))

Theoremsbbib 2369* Reversal of substitution. (Contributed by AV, 6-Aug-2023.) (Proof shortened by Wolf Lammen, 4-Sep-2023.)
𝑦𝜑    &   𝑥𝜓       (∀𝑦([𝑦 / 𝑥]𝜑𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓))

Theoremsbbibvv 2370* Reversal of substitution. (Contributed by AV, 6-Aug-2023.)
(∀𝑦([𝑦 / 𝑥]𝜑𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓))

TheoremcleljustALT 2371* Alternate proof of cleljust 2120. It is kept here and should not be modified because it is referenced on the Metamath Proof Explorer Home Page (mmset.html) as an example of how disjoint variable conditions are inherited by substitutions. (Contributed by NM, 28-Jan-2004.) (Revised by BJ, 29-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))

TheoremcleljustALT2 2372* Alternate proof of cleljust 2120. Compared with cleljustALT 2371, it uses nfv 1915 followed by equsexv 2266 instead of ax-5 1911 followed by equsexhv 2296, so it uses the idiom 𝑥𝜑 instead of 𝜑 → ∀𝑥𝜑 to express non-freeness. This style is generally preferred for later theorems. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.) (Revised by BJ, 29-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))

Theoremequs5aALT 2373 Alternate proof of equs5a 2469. Uses ax-12 2175 but not ax-13 2379. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))

Theoremequs5eALT 2374 Alternate proof of equs5e 2470. Uses ax-12 2175 but not ax-13 2379. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Theoremaxc11r 2375 Same as axc11 2441 but with reversed antecedent. Note the use of ax-12 2175 (and not merely ax12v 2176 as in axc11rv 2263).

This theorem is mostly used to eliminate conditions requiring set variables be distinct (cf. cbvaev 2058 and aecom 2438, for example) in proofs. In practice, theorems beyond elementary set theory do not really benefit from such eliminations. As of 2024, it is used in conjunction with ax-13 2379 only, and like that, it should be applied only in niches where indispensable. (Contributed by NM, 25-Jul-2015.)

(∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))

Theoremdral1v 2376* Formula-building lemma for use with the Distinctor Reduction Theorem. Version of dral1 2450 with a disjoint variable condition, which does not require ax-13 2379. Remark: the corresponding versions for dral2 2449 and drex2 2453 are instances of albidv 1921 and exbidv 1922 respectively. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2176. (Revised by Wolf Lammen, 30-Mar-2024.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Theoremdrex1v 2377* Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drex1 2452 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 27-Feb-2005.) (Revised by BJ, 17-Jun-2019.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))

Theoremdrnf1v 2378* Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drnf1 2454 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by Mario Carneiro, 4-Oct-2016.) (Revised by BJ, 17-Jun-2019.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))

1.5.4  Axiom scheme ax-13 (Quantified Equality)

Axiomax-13 2379 Axiom of Quantified Equality. One of the equality and substitution axioms of predicate calculus with equality.

An equivalent way to express this axiom that may be easier to understand is 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥𝑦 = 𝑧))) (see ax13b 2039). Recall that in the intended interpretation, our variables are metavariables ranging over the variables of predicate calculus (the object language). In order for the first antecedent ¬ 𝑥 = 𝑦 to hold, 𝑥 and 𝑦 must have different values and thus cannot be the same object-language variable (so they are effectively "distinct variables" even though no \$d is present). Similarly, 𝑥 and 𝑧 cannot be the same object-language variable. Therefore, 𝑥 will not occur in the wff 𝑦 = 𝑧 when the first two antecedents hold, so analogous to ax-5 1911, the conclusion (𝑦 = 𝑧 → ∀𝑥𝑦 = 𝑧) follows. Note that ax-5 1911 cannot prove this because its distinct variable (\$d) requirement is not satisfied directly but only indirectly (outside of Metamath) by the argument above.

The original version of this axiom was ax-c9 36205 and was replaced with this shorter ax-13 2379 in December 2015. The old axiom is proved from this one as theorem axc9 2389.

The primary purpose of this axiom is to provide a way to introduce the quantifier 𝑥 on 𝑦 = 𝑧 even when 𝑥 and 𝑦 are substituted with the same variable. In this case, the first antecedent becomes ¬ 𝑥 = 𝑥 and the axiom still holds.

This axiom is mostly used to eliminate conditions requiring set variables be distinct (cf. ax6ev 1972 and ax6e 2390, for example) in proofs. In practice, theorems beyond elementary set theory do not really benefit from such eliminations, so direct or indirect application of this axiom is discouraged now. You need to explicitly confirm its use in case you see a sensible application in a niche.

After some assisting contributions by others over the years, it was in particular the extensive work of Gino Giotto in 2024 that helped reducing dependencies on this axiom on a large scale.

Although this version is shorter, the original version axc9 2389 may be more practical to work with because of the "distinctor" form of its antecedents. A typical application of axc9 2389 is in dvelimh 2461 which converts a distinct variable pair to the distinctor antecedent ¬ ∀𝑥𝑥 = 𝑦. In particular, it is conjectured that it is not possible to prove ax6 2391 from ax6v 1971 without this axiom.

This axiom can be weakened if desired by adding distinct variable restrictions on pairs 𝑥, 𝑧 and 𝑦, 𝑧. To show that, we add these restrictions to theorem ax13v 2380 and use only ax13v 2380 for further derivations. Thus, ax13v 2380 should be the only theorem referencing this axiom. Other theorems can reference either ax13v 2380 (preferred) or ax13 2382 (if the stronger form is needed).

This axiom scheme is logically redundant (see ax13w 2137) but is used as an auxiliary axiom scheme to achieve scheme completeness (i.e. so that all possible cases of bundling can be proved; see text linked at mmtheorems.html#ax6dgen 2137). It is not known whether this axiom can be derived from the others. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.)

𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremax13v 2380* A weaker version of ax-13 2379 with distinct variable restrictions on pairs 𝑥, 𝑧 and 𝑦, 𝑧. In order to show (with ax13 2382) that this weakening is still adequate, this should be the only theorem referencing ax-13 2379 directly.

Had we additionally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1911. So essentially this theorem states, that a distinct variable condition can be replaced with an inequality between set variables. Preferably, use the version ax13w 2137 to avoid the propagation of ax-13 2379. (Contributed by NM, 30-Jun-2016.) (New usage is discouraged.)

𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremax13lem1 2381* A version of ax13v 2380 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. The proof of ax13 2382 bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)
𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremax13 2382 Derive ax-13 2379 from ax13v 2380 and Tarski's FOL. This shows that the weakening in ax13v 2380 is still sufficient for a complete system. Preferably, use the weaker ax13w 2137 to avoid the propagation of ax-13 2379. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.) Reduce axiom usage (Revised by Wolf Lammen, 2-Jun-2021.) (New usage is discouraged.)
𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremax13lem2 2383* Lemma for nfeqf2 2384. This lemma is equivalent to ax13v 2380 with one distinct variable constraint removed. (Contributed by Wolf Lammen, 8-Sep-2018.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Oct-2020.) (New usage is discouraged.)
𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))

Theoremnfeqf2 2384* An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by Wolf Lammen, 9-Jun-2019.) Remove dependency on ax-12 2175. (Revised by Wolf Lammen, 16-Dec-2022.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)

Theoremdveeq2 2385* Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.) Remove dependency on ax-11 2158. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremnfeqf1 2386* An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by Wolf Lammen, 10-Jun-2019.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)

Theoremdveeq1 2387* Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 2-Jan-2002.) Remove dependency on ax-11 2158. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremnfeqf 2388 A variable is effectively not free in an equality if it is not either of the involved variables. version of ax-c9 36205. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 2158. (Revised by Wolf Lammen, 6-Sep-2018.) (New usage is discouraged.)
((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)

Theoremaxc9 2389 Derive set.mm's original ax-c9 36205 from the shorter ax-13 2379. Usage is discouraged to avoid uninformed ax-13 2379 propagation. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 29-Apr-2018.) (New usage is discouraged.)
(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Theoremax6e 2390 At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-4 1811 through ax-9 2121, all axioms other than ax-6 1970 are believed to be theorems of free logic, although the system without ax-6 1970 is not complete in free logic.

Usage of this theorem is discouraged because it depends on ax-13 2379. It is preferred to use ax6ev 1972 when it is sufficient. (Contributed by NM, 14-May-1993.) Shortened after ax13lem1 2381 became available. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)

𝑥 𝑥 = 𝑦

Theoremax6 2391 Theorem showing that ax-6 1970 follows from the weaker version ax6v 1971. (Even though this theorem depends on ax-6 1970, all references of ax-6 1970 are made via ax6v 1971. An earlier version stated ax6v 1971 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-6 1970 so that all proofs can be traced back to ax6v 1971. Usage of this theorem is discouraged because it depends on ax-13 2379. When possible, use the weaker ax6v 1971 rather than ax6 2391 since the ax6v 1971 derivation is much shorter and requires fewer axioms. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 4-Feb-2018.) (New usage is discouraged.)

¬ ∀𝑥 ¬ 𝑥 = 𝑦

Theoremaxc10 2392 Show that the original axiom ax-c10 36201 can be derived from ax6 2391 and axc7 2325 (on top of propositional calculus, ax-gen 1797, and ax-4 1811). See ax6fromc10 36211 for the rederivation of ax6 2391 from ax-c10 36201.

Normally, axc10 2392 should be used rather than ax-c10 36201, except by theorems specifically studying the latter's properties. Usage of this theorem has been discouraged later on to avoid ax-13 2379 propagation. Check out bj-axc10v 34249 for a weaker version requiring less axioms. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Theoremspimt 2393 Closed theorem form of spim 2394. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 21-Mar-2023.) (New usage is discouraged.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))

Theoremspim 2394 Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2394 series of theorems requires that only one direction of the substitution hypothesis hold. Usage of this theorem is discouraged because it depends on ax-13 2379. Check out spimw 1973 for a version requiring less axioms. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspimed 2395 Deduction version of spime 2396. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker spimedv 2195 if possible. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.) (New usage is discouraged.)
(𝜒 → Ⅎ𝑥𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜒 → (𝜑 → ∃𝑥𝜓))

Theoremspime 2396 Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. Usage of this theorem is discouraged because it depends on ax-13 2379. Check out spimew 1974 for a weaker version requiring less axioms. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) (New usage is discouraged.)
𝑥𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theoremspimv 2397* A version of spim 2394 with a distinct variable requirement instead of a bound-variable hypothesis. Usage of this theorem is discouraged because it depends on ax-13 2379. See spimfv 2239 and spimvw 2002 for versions requiring fewer axioms. (Contributed by NM, 31-Jul-1993.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

TheoremspimvALT 2398* Alternate proof of spimv 2397. Note that it requires only ax-1 6 through ax-5 1911 together with ax6e 2390. Currently, proofs derive from ax6v 1971, but if ax-6 1970 could be used instead, this proof would reduce axiom usage. (Contributed by NM, 31-Jul-1993.) Remove dependency on ax-10 2142. (Revised by BJ, 29-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspimev 2399* Distinct-variable version of spime 2396. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker spimevw 2001 if possible. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theoremspv 2400* Specialization, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker spvv 2003 if possible. (Contributed by NM, 30-Aug-1993.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

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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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 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