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Theorem List for Metamath Proof Explorer - 2001-2100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremexsbim 2001* One direction of the equivalence in exsb 2373 is based on fewer axioms. (Contributed by Wolf Lammen, 2-Mar-2023.)
(∃𝑦𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)

Theoremequsv 2002* If a formula does not contain a variable 𝑥, then it is equivalent to the corresponding prototype of substitution with a fresh variable (see sb6 2086). (Contributed by BJ, 23-Jul-2023.)
(∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑)

Theoremequsalvw 2003* Version of equsalv 2261 with a disjoint variable condition, and of equsal 2435 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsexvw 2004. (Contributed by BJ, 31-May-2019.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsexvw 2004* Version of equsexv 2262 with a disjoint variable condition, and of equsex 2436 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsalvw 2003. (Contributed by BJ, 31-May-2019.) (Proof shortened by Wolf Lammen, 23-Oct-2023.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

TheoremequsexvwOLD 2005* Obsolete version of equsexvw 2004 as of 23-Oct-2023. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremcbvaliw 2006* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.)
(∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜓 → ∀𝑥 ¬ 𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremcbvalivw 2007* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

1.4.7  Axiom scheme ax-7 (Equality)

Axiomax-7 2008 Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. It states that equality is a right-Euclidean binary relation (this is similar, but not identical, to being transitive, which is proved as equtr 2021). This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom C7 of [Monk2] p. 105 and Axiom Scheme C8' in [Megill] p. 448 (p. 16 of the preprint).

The equality symbol was invented in 1557 by Robert Recorde. He chose a pair of parallel lines of the same length because "noe .2. thynges, can be moare equalle".

We prove in ax7 2016 that this axiom can be recovered from its weakened version ax7v 2009 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-7 2008 should be ax7v 2009. See the comment of ax7v 2009 for more details on these matters. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 7-Dec-2020.) Use ax7 2016 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremax7v 2009* Weakened version of ax-7 2008, with a disjoint variable condition on 𝑥, 𝑦. This should be the only proof referencing ax-7 2008, and it should be referenced only by its two weakened versions ax7v1 2010 and ax7v2 2011, from which ax-7 2008 is then rederived as ax7 2016, which shows that either ax7v 2009 or the conjunction of ax7v1 2010 and ax7v2 2011 is sufficient.

In ax7v 2009, it is still allowed to substitute the same variable for 𝑥 and 𝑧, or the same variable for 𝑦 and 𝑧. Therefore, ax7v 2009 "bundles" (a term coined by Raph Levien) its "principal instance" (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧)) with 𝑥, 𝑦, 𝑧 distinct, and its "degenerate instances" (𝑥 = 𝑦 → (𝑥 = 𝑥𝑦 = 𝑥)) and (𝑥 = 𝑦 → (𝑥 = 𝑦𝑦 = 𝑦)) with 𝑥, 𝑦 distinct. These degenerate instances are for instance used in the proofs of equcomiv 2014 and equid 2012 respectively. (Contributed by BJ, 7-Dec-2020.) Use ax7 2016 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremax7v1 2010* First of two weakened versions of ax7v 2009, with an extra disjoint variable condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremax7v2 2011* Second of two weakened versions of ax7v 2009, with an extra disjoint variable condition on 𝑦, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremequid 2012 Identity law for equality. Lemma 2 of [KalishMontague] p. 85. See also Lemma 6 of [Tarski] p. 68. (Contributed by NM, 1-Apr-2005.) (Revised by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 22-Aug-2020.)
𝑥 = 𝑥

Theoremnfequid 2013 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. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)
𝑦 𝑥 = 𝑥

Theoremequcomiv 2014* Weaker form of equcomi 2017 with a disjoint variable condition on 𝑥, 𝑦. This is an intermediate step and equcomi 2017 is fully recovered later. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremax6evr 2015* A commuted form of ax6ev 1965. (Contributed by BJ, 7-Dec-2020.)
𝑥 𝑦 = 𝑥

Theoremax7 2016 Proof of ax-7 2008 from ax7v1 2010 and ax7v2 2011 (and earlier axioms), proving sufficiency of the conjunction of the latter two weakened versions of ax7v 2009, which is itself a weakened version of ax-7 2008.

Note that the weakened version of ax-7 2008 obtained by adding a disjoint variable condition on 𝑥, 𝑧 (resp. on 𝑦, 𝑧) does not permit, together with the other axioms, to prove reflexivity (resp. symmetry). (Contributed by BJ, 7-Dec-2020.)

(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremequcomi 2017 Commutative law for equality. Equality is a symmetric relation. Lemma 3 of [KalishMontague] p. 85. See also Lemma 7 of [Tarski] p. 69. (Contributed by NM, 10-Jan-1993.) (Revised by NM, 9-Apr-2017.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremequcom 2018 Commutative law for equality. Equality is a symmetric relation. (Contributed by NM, 20-Aug-1993.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremequcomd 2019 Deduction form of equcom 2018, symmetry of equality. For the versions for classes, see eqcom 2832 and eqcomd 2831. (Contributed by BJ, 6-Oct-2019.)
(𝜑𝑥 = 𝑦)       (𝜑𝑦 = 𝑥)

Theoremequcoms 2020 An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 10-Jan-1993.)
(𝑥 = 𝑦𝜑)       (𝑦 = 𝑥𝜑)

Theoremequtr 2021 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
(𝑥 = 𝑦 → (𝑦 = 𝑧𝑥 = 𝑧))

Theoremequtrr 2022 A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)
(𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))

Theoremequeuclr 2023 Commuted version of equeucl 2024 (equality is left-Euclidean). (Contributed by BJ, 12-Apr-2021.)
(𝑥 = 𝑧 → (𝑦 = 𝑧𝑦 = 𝑥))

Theoremequeucl 2024 Equality is a left-Euclidean binary relation. (Right-Euclideanness is stated in ax-7 2008.) Curried (exported) form of equtr2 2027. (Contributed by BJ, 11-Apr-2021.)
(𝑥 = 𝑧 → (𝑦 = 𝑧𝑥 = 𝑦))

Theoremequequ1 2025 An equivalence law for equality. (Contributed by NM, 1-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremequequ2 2026 An equivalence law for equality. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.) (Proof shortened by BJ, 12-Apr-2021.)
(𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))

Theoremequtr2 2027 Equality is a left-Euclidean binary relation. Uncurried (imported) form of equeucl 2024. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by BJ, 11-Apr-2021.)
((𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)

Theoremstdpc6 2028 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 2245.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)
𝑥 𝑥 = 𝑥

Theoremequvinv 2029* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2138, ax-13 2385. (Revised by Wolf Lammen, 10-Jun-2019.) Move the quantified variable (𝑧) to the left of the equality signs. (Revised by Wolf Lammen, 11-Apr-2021.) (Proof shortened by Wolf Lammen, 12-Jul-2022.)
(𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))

Theoremequvinva 2030* A modified version of the forward implication of equvinv 2029 adapted to common usage. (Contributed by Wolf Lammen, 8-Sep-2018.)
(𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑦 = 𝑧))

Theoremequvelv 2031* A biconditional form of equvel 2476 with disjoint variable conditions and proved from Tarski's FOL axiom schemes. (Contributed by Andrew Salmon, 2-Jun-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Apr-2021.) (Proof shortened by Wolf Lammen, 12-Jul-2022.)
(∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ↔ 𝑥 = 𝑦)

Theoremax13b 2032 An equivalence between two ways of expressing ax-13 2385. See the comment for ax-13 2385. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.) (Revised by BJ, 15-Sep-2020.)
((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧𝜑)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧𝜑))))

Theoremspfw 2033* Weak version of sp 2174. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 10-Oct-2021.)
𝜓 → ∀𝑥 ¬ 𝜓)    &   (∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜑 → ∀𝑦 ¬ 𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜑)

Theoremspw 2034* Weak version of the specialization scheme sp 2174. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2174 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2174 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2132 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2174 are spfw 2033 (minimal distinct variable requirements), spnfw 1977 (when 𝑥 is not free in ¬ 𝜑), spvw 1978 (when 𝑥 does not appear in 𝜑), sptruw 1800 (when 𝜑 is true), and spfalw 1997 (when 𝜑 is false). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜑)

Theoremcbvalw 2035* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
(∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜓 → ∀𝑥 ¬ 𝜓)    &   (∀𝑦𝜓 → ∀𝑥𝑦𝜓)    &   𝜑 → ∀𝑦 ¬ 𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremcbvalvw 2036* Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvalv 2414 for a version with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremcbvexvw 2037* Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvexv 2415 for a version with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 19-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theoremcbvaldvaw 2038* Version of cbvaldva 2426 with a disjoint variable condition, requiring fewer axioms. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage, along an idea of Gino Giotto. (Revised by Wolf Lammen, 10-Feb-2024.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbvexdvaw 2039* Version of cbvexdva 2427 with a disjoint variable condition, requiring fewer axioms. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Feb-2024.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Theoremcbval2vw 2040* Version of cbval2vv 2431 with more disjoint variable conditions, which requires fewer axioms . (Contributed by Gino Giotto, 10-Jan-2024.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theoremcbvex2vw 2041* Version of cbvex2vv 2432 with more disjoint variable conditions, which requires fewer axioms . (Contributed by Gino Giotto, 10-Jan-2024.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theoremcbvex4vw 2042* Version of cbvex4v 2433 with more disjoint variable conditions, which requires fewer axioms. (Contributed by Gino Giotto, 10-Jan-2024.)
((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))    &   ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))       (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)

Theoremalcomiw 2043* Weak version of alcom 2155. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Dec-2023.)
(𝑦 = 𝑧 → (𝜑𝜓))       (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

TheoremalcomiwOLD 2044* Obsolete version of alcomiw 2043 as of 28-Dec-2023. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 12-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑦 = 𝑧 → (𝜑𝜓))       (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremhbn1fw 2045* Weak version of ax-10 2138 from which we can prove any ax-10 2138 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
(∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜓 → ∀𝑥 ¬ 𝜓)    &   (∀𝑦𝜓 → ∀𝑥𝑦𝜓)    &   𝜑 → ∀𝑦 ¬ 𝜑)    &   (¬ ∀𝑦𝜓 → ∀𝑥 ¬ ∀𝑦𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremhbn1w 2046* Weak version of hbn1 2139. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremhba1w 2047* Weak version of hba1 2295. See comments for ax10w 2126. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 10-Oct-2021.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑥𝑥𝜑)

Theoremhbe1w 2048* Weak version of hbe1 2140. See comments for ax10w 2126. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 → ∀𝑥𝑥𝜑)

Theoremhbalw 2049* Weak version of hbal 2166. Uses only Tarski's FOL axiom schemes. Unlike hbal 2166, this theorem requires that 𝑥 and 𝑦 be distinct, i.e. not be bundled. (Contributed by NM, 19-Apr-2017.)
(𝑥 = 𝑧 → (𝜑𝜓))    &   (𝜑 → ∀𝑥𝜑)       (∀𝑦𝜑 → ∀𝑥𝑦𝜑)

Theoremspaev 2050* A special instance of sp 2174 applied to an equality with a disjoint variable condition. Unlike the more general sp 2174, we can prove this without ax-12 2169. Instance of aeveq 2054.

The antecedent 𝑥𝑥 = 𝑦 with distinct 𝑥 and 𝑦 is a characteristic of a degenerate universe, in which just one object exists. Actually more than one object may still exist, but if so, we give up on equality as a discriminating term.

Separating this degenerate case from a richer universe, where inequality is possible, is a common proof idea. The name of this theorem follows a convention, where the condition 𝑥𝑥 = 𝑦 is denoted by 'aev', a shorthand for 'all equal, with a distinct variable condition'. (Contributed by Wolf Lammen, 14-Mar-2021.)

(∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)

Theoremcbvaev 2051* Change bound variable in an equality with a disjoint variable condition. Instance of aev 2055. (Contributed by NM, 22-Jul-2015.) (Revised by BJ, 18-Jun-2019.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦)

Theoremaevlem0 2052* Lemma for aevlem 2053. Instance of aev 2055. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-12 2169. (Revised by Wolf Lammen, 14-Mar-2021.) (Revised by BJ, 29-Mar-2021.) (Proof shortened by Wolf Lammen, 30-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥)

Theoremaevlem 2053* Lemma for aev 2055 and axc16g 2254. Change free and bound variables. Instance of aev 2055. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2385, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 29-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑡)

Theoremaeveq 2054* The antecedent 𝑥𝑥 = 𝑦 with a disjoint variable condition (typical of a one-object universe) forces equality of everything. (Contributed by Wolf Lammen, 19-Mar-2021.)
(∀𝑥 𝑥 = 𝑦𝑧 = 𝑡)

Theoremaev 2055* A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 2153. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2385, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 2169. (Revised by Wolf Lammen, 19-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑡 = 𝑢)

Theoremaev2 2056* A version of aev 2055 with two universal quantifiers in the consequent. One can prove similar statements with arbitrary numbers of universal quantifiers in the consequent (the series begins with aeveq 2054, aev 2055, aev2 2056).

Using aev 2055 and alrimiv 1921, one can actually prove (with no more axioms) any scheme of the form (∀𝑥𝑥 = 𝑦 PHI) , DV (𝑥, 𝑦) where PHI involves only setvar variables and the connectors , , , , , =, , , ∃*, ∃!, . An example is given by aevdemo 28153. This list cannot be extended to ¬ or since the scheme 𝑥𝑥 = 𝑦 is consistent with ax-mp 5, ax-gen 1789, ax-1 6-- ax-13 2385 (as the one-element universe shows).

(Contributed by BJ, 23-Mar-2021.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑡 𝑢 = 𝑣)

Theoremhbaev 2057* Version of hbae 2450 with a disjoint variable condition, requiring fewer axioms. Instance of aev2 2056. (Contributed by Wolf Lammen, 22-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)

Theoremnaev 2058* If some set variables can assume different values, then any two distinct set variables cannot always be the same. (Contributed by Wolf Lammen, 10-Aug-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑢 𝑢 = 𝑣)

Theoremnaev2 2059* Generalization of hbnaev 2060. (Contributed by Wolf Lammen, 9-Apr-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑡 𝑡 = 𝑢)

Theoremhbnaev 2060* Any variable is free in ¬ ∀𝑥𝑥 = 𝑦, if 𝑥 and 𝑦 are distinct. This condition is dropped in hbnae 2451, at the expense of more axiom dependencies. Instance of naev2 2059. (Contributed by Wolf Lammen, 9-Apr-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

1.4.8  Define proper substitution

Theoremsbjust 2061* Justification theorem for df-sb 2063 proved from Tarski's FOL. (Contributed by BJ, 22-Jan-2023.)
(∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧𝜑)))

Syntaxwsb 2062 Extend wff definition to include proper substitution (read "the wff that results when 𝑦 is properly substituted for 𝑥 in wff 𝜑"). (Contributed by NM, 24-Jan-2006.)
wff [𝑦 / 𝑥]𝜑

Definitiondf-sb 2063* Define proper substitution. For our notation, we use [𝑡 / 𝑥]𝜑 to mean "the wff that results from the proper substitution of 𝑡 for 𝑥 in the wff 𝜑". That is, 𝑡 properly replaces 𝑥. For example, [𝑡 / 𝑥]𝑧𝑥 is the same as 𝑧𝑡 (when 𝑥 and 𝑧 are distinct), as shown in elsb4 2123.

Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, "𝜑(𝑡) is the wff that results when 𝑡 is properly substituted for 𝑥 in 𝜑(𝑥)". For example, if the original 𝜑(𝑥) is 𝑥 = 𝑡, then 𝜑(𝑡) is 𝑡 = 𝑡, from which we obtain that 𝜑(𝑥) is 𝑥 = 𝑥. So what exactly does 𝜑(𝑥) mean? Curry's notation solves this problem.

A very similar notation, namely (𝑦𝑥)𝜑, was introduced in Bourbaki's Set Theory (Chapter 1, Description of Formal Mathematic, 1953).

In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 2083, sbcom2 2160 and sbid2v 2549).

Note that our definition is valid even when 𝑥 and 𝑡 are replaced with the same variable, as sbid 2250 shows. We achieve this by applying twice Tarski's definition sb6 2086 which is valid for disjoint variables, and introducing a dummy variable 𝑦 which isolates 𝑥 from 𝑡, as in dfsb7 2279 with respect to sb5 2270. We can also achieve this by having 𝑥 free in the first conjunct and bound in the second, as the alternate definition dfsb1 2508 shows. Another version that mixes free and bound variables is dfsb3 2531. When 𝑥 and 𝑡 are distinct, we can express proper substitution with the simpler expressions of sb5 2270 and sb6 2086.

Note that the occurrences of a given variable in the definiens are either all bound (𝑥, 𝑦) or all free (𝑡). Also note that the definiens uses only primitive symbols.

This double level definition will make several proofs using it appear as doubled. Alternately, one could often first prove as a lemma the same theorem with a disjoint variable condition on the substitute and the substituted variables, and then prove the original theorem by applying this lemma twice in a row. (Contributed by NM, 10-May-1993.) Revised from the original definition dfsb1 2508. (Revised by BJ, 22-Dec-2020.)

([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremsbt 2064 A substitution into a theorem yields a theorem. See sbtALT 2067 for a shorter proof requiring more axioms. See chvar 2409 and chvarv 2410 for versions using implicit substitution. (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Jul-2018.) Revise df-sb 2063. (Revised by Steven Nguyen, 6-Jul-2023.)
𝜑       [𝑡 / 𝑥]𝜑

Theoremsbtru 2065 The result of substituting in the truth constant "true" is true. (Contributed by BJ, 2-Sep-2023.)
[𝑦 / 𝑥]⊤

Theoremstdpc4 2066 The specialization axiom of standard predicate calculus. It states that if a statement 𝜑 holds for all 𝑥, then it also holds for the specific case of 𝑡 (properly) substituted for 𝑥. Translated to traditional notation, it can be read: "𝑥𝜑(𝑥) → 𝜑(𝑡), provided that 𝑡 is free for 𝑥 in 𝜑(𝑥)". Axiom 4 of [Mendelson] p. 69. See also spsbc 3788 and rspsbc 3865. (Contributed by NM, 14-May-1993.) Revise df-sb 2063. (Revised by BJ, 22-Dec-2020.)
(∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)

TheoremsbtALT 2067 Alternate proof of sbt 2064, shorter but using ax-4 1803 and ax-5 1904. (Contributed by NM, 21-Jan-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑       [𝑦 / 𝑥]𝜑

Theorem2stdpc4 2068 A double specialization using explicit substitution. This is Theorem PM*11.1 in [WhiteheadRussell] p. 159. See stdpc4 2066 for the analogous single specialization. See 2sp 2177 for another double specialization. (Contributed by Andrew Salmon, 24-May-2011.) (Revised by BJ, 21-Oct-2018.)
(∀𝑥𝑦𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

Theoremsbi1 2069 Distribute substitution over implication. (Contributed by NM, 14-May-1993.) Remove dependencies on axioms. (Revised by Steven Nguyen, 24-Jul-2023.)
([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Theoremspsbim 2070 Distribute substitution over implication. Closed form of sbimi 2072. Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) Revise df-sb 2063. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 24-Jul-2023.)
(∀𝑥(𝜑𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))

Theoremspsbbi 2071 Biconditional property for substitution. Closed form of sbbii 2074. Specialization of biconditional. (Contributed by NM, 2-Jun-1993.) Revise df-sb 2063. (Revised by BJ, 22-Dec-2020.)
(∀𝑥(𝜑𝜓) → ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓))

Theoremsbimi 2072 Distribute substitution over implication. (Contributed by NM, 25-Jun-1998.) Revise df-sb 2063. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 24-Jul-2023.)
(𝜑𝜓)       ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓)

Theoremsb2imi 2073 Distribute substitution over implication. Compare al2imi 1809. (Contributed by Steven Nguyen, 13-Aug-2023.)
(𝜑 → (𝜓𝜒))       ([𝑡 / 𝑥]𝜑 → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒))

Theoremsbbii 2074 Infer substitution into both sides of a logical equivalence. (Contributed by NM, 14-May-1993.)
(𝜑𝜓)       ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓)

Theorem2sbbii 2075 Infer double substitution into both sides of a logical equivalence. (Contributed by AV, 30-Jul-2023.)
(𝜑𝜓)       ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)

Theoremsbimdv 2076* Deduction substituting both sides of an implication, with 𝜑 and 𝑥 disjoint. See also sbimd 2238. (Contributed by Wolf Lammen, 6-May-2023.) Revise df-sb 2063. (Revised by Steven Nguyen, 6-Jul-2023.)
(𝜑 → (𝜓𝜒))       (𝜑 → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒))

Theoremsbbidv 2077* Deduction substituting both sides of a biconditional, with 𝜑 and 𝑥 disjoint. See also sbbid 2239. (Contributed by Wolf Lammen, 6-May-2023.) (Proof shortened by Steven Nguyen, 6-Jul-2023.)
(𝜑 → (𝜓𝜒))       (𝜑 → ([𝑡 / 𝑥]𝜓 ↔ [𝑡 / 𝑥]𝜒))

TheoremsbbidvOLD 2078* Obsolete version of sbbidv 2077 as of 6-Jul-2023. Deduction substituting both sides of a biconditional, with 𝜑 and 𝑥 disjoint. See also sbbid 2239. (Contributed by Wolf Lammen, 6-May-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))

Theoremsban 2079 Conjunction inside and outside of a substitution are equivalent. Compare 19.26 1864. (Contributed by NM, 14-May-1993.) (Proof shortened by Steven Nguyen, 13-Aug-2023.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))

Theoremsb3an 2080 Threefold conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.)
([𝑦 / 𝑥](𝜑𝜓𝜒) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓 ∧ [𝑦 / 𝑥]𝜒))

Theoremspsbe 2081 Existential generalization: if a proposition is true for a specific instance, then there exists an instance where it is true. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.) Revise df-sb 2063. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 11-Jul-2023.)
([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)

TheoremspsbeOLD 2082 Obsolete version of spsbe 2081 as of 11-Jul-2023. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.) Revise df-sb 2063. (Revised by BJ, 22-Dec-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)

Theoremsbequ 2083 Equality property for substitution, from Tarski's system. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 14-May-1993.) Revise df-sb 2063. (Revised by BJ, 30-Dec-2020.)
(𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))

Theoremsbequi 2084 An equality theorem for substitution. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 15-Sep-2018.) (Proof shortened by Steven Nguyen, 7-Jul-2023.)
(𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))

TheoremsbequOLD 2085 Obsolete proof of sbequ 2083 as of 7-Jul-2023. An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))

Theoremsb6 2086* Alternate definition of substitution when variables are disjoint. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. The implication "to the left" also holds without a disjoint variable condition (sb2 2502). Theorem sb6f 2535 replaces the disjoint variable condition with a non-freeness hypothesis. Theorem sb4b 2497 replaces it with a distinctor antecedent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) Revise df-sb 2063. (Revised by BJ, 22-Dec-2020.) Remove use of ax-11 2153. (Revised by Steven Nguyen, 7-Jul-2023.) (Proof shortened by Wolf Lammen, 16-Jul-2023.)
([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑))

Theorem2sb6 2087* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))

Theoremsb1v 2088* One direction of sb5 2270, provable from fewer axioms. Version of sb1 2501 with a disjoint variable condition using fewer axioms. (Contributed by Wolf Lammen, 20-Jan-2024.)
([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))

Theoremsb4vOLD 2089* Obsolete as of 30-Jul-2023. Use sb6 2086 instead. (Contributed by BJ, 23-Jun-2019.) (Proof shortened by Steven Nguyen, 8-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsb2vOLD 2090* Obsolete as of 30-Jul-2023. Use sb6 2086 instead. Version of sb2 2502 with a disjoint variable condition, which does not require ax-13 2385. (Contributed by BJ, 31-May-2019.) Revise df-sb 2063. (Revised by Steven Nguyen, 8-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)

Theoremsbv 2091* Substitution for a variable not occurring in a proposition. See sbf 2264 for a version without disjoint variable condition on 𝑥, 𝜑. If one adds a disjoint variable condition on 𝑥, 𝑡, then sbv 2091 can be proved directly by chaining equsv 2002 with sb6 2086. (Contributed by BJ, 22-Dec-2020.)
([𝑡 / 𝑥]𝜑𝜑)

Theoremsbcom4 2092* Commutativity law for substitution. This theorem was incorrectly used as our previous version of pm11.07 2093 but may still be useful. (Contributed by Andrew Salmon, 17-Jun-2011.) (Proof shortened by Jim Kingdon, 22-Jan-2018.)
([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

Theorempm11.07 2093 Axiom *11.07 in [WhiteheadRussell] p. 159. The original reads: *11.07 "Whatever possible argument 𝑥 may be, 𝜑(𝑥, 𝑦) is true whatever possible argument 𝑦 may be" implies the corresponding statement with 𝑥 and 𝑦 interchanged except in "𝜑(𝑥, 𝑦)". Under our formalism this appears to correspond to idi 1 and not to sbcom4 2092 as earlier thought. See https://groups.google.com/g/metamath/c/iS0fOvSemC8/m/M1zTH8wxCAAJ 2092. (Contributed by BJ, 16-Sep-2018.) (New usage is discouraged.)
𝜑       𝜑

Theoremsbrimvlem 2094* Common proof template for sbrimvw 2095 and sbrimv 2308. The hypothesis is an instance of 19.21 2200. (Contributed by Wolf Lammen, 29-Jan-2024.)
(∀𝑥(𝜑 → (𝑥 = 𝑦𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓)))       ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

Theoremsbrimvw 2095* Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim 2307 and sbrimv 2308 based on fewer axioms, but with more disjoint variable conditions. Based on an idea of Gino Giotto. (Contributed by Wolf Lammen, 29-Jan-2024.)
([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

Theoremsbievw 2096* Version of sbie 2542 and sbiev 2324 with more disjoint variable conditions, requiring fewer axioms. (Contributed by BJ, 18-Jul-2023.)
(𝑥 = 𝑦 → (𝜑𝜓))       ([𝑦 / 𝑥]𝜑𝜓)

Theoremsbiedvw 2097* Version of sbied 2543 and sbiedv 2544 with more disjoint variable conditions, requiring fewer axioms. (Contributed by Gino Giotto, 29-Jan-2024.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))

Theorem2sbievw 2098* Version of 2sbiev 2545 with more disjoint variable conditions, requiring fewer axioms. (Contributed by Gino Giotto, 10-Jan-2024.)
((𝑥 = 𝑡𝑦 = 𝑢) → (𝜑𝜓))       ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑𝜓)

Theoremsbcom3vv 2099* Version of sbcom3 2546 with a disjoint variable condition using fewer axioms. (Contributed by BJ, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 19-Jan-2023.)
([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑)

Theoremsbievw2 2100* sbievw 2096 applied twice, avoiding a DV condition on 𝑥, 𝑦. Based on proofs by Wolf Lammen. (Contributed by Steven Nguyen, 29-Jul-2023.)
(𝑥 = 𝑤 → (𝜑𝜒))    &   (𝑤 = 𝑦 → (𝜒𝜓))       ([𝑦 / 𝑥]𝜑𝜓)

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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 44701-44734
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