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Theorem List for Metamath Proof Explorer - 34601-34700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremwl-luk-pm2.21 34601 From a wff and its negation, anything follows. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. Copy of pm2.21 123 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑 → (𝜑𝜓))
 
Theoremwl-luk-con1i 34602 A contraposition inference. Copy of con1i 149 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑𝜓)       𝜓𝜑)
 
Theoremwl-luk-ja 34603 Inference joining the antecedents of two premises. Copy of ja 187 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑𝜒)    &   (𝜓𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremwl-luk-imim2 34604 A closed form of syllogism (see syl 17). Theorem *2.05 of [WhiteheadRussell] p. 100. Copy of imim2 58 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))
 
Theoremwl-luk-a1d 34605 Deduction introducing an embedded antecedent. Copy of imim2 58 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓))
 
Theoremwl-luk-ax2 34606 ax-2 7 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
 
Theoremwl-luk-id 34607 Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. Copy of id 22 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜑)
 
Theoremwl-luk-notnotr 34608 Converse of double negation. Theorem *2.14 of [WhiteheadRussell] p. 102. In classical logic (our logic) this is always true. In intuitionistic logic this is not always true; in intuitionistic logic, when this is true for some 𝜑, then 𝜑 is stable. Copy of notnotr 132 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(¬ ¬ 𝜑𝜑)
 
Theoremwl-luk-pm2.04 34609 Swap antecedents. Theorem *2.04 of [WhiteheadRussell] p. 100. This was the third axiom in Frege's logic system, specifically Proposition 8 of [Frege1879] p. 35. Copy of pm2.04 90 with a different proof. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))
 
20.18.2  Implication chains
 
Theoremwl-section-impchain 34610 An implication like (𝜓𝜑) with one antecedent can easily be extended by prepending more and more antecedents, as in (𝜒 → (𝜓𝜑)) or (𝜃 → (𝜒 → (𝜓𝜑))). I call these expressions implication chains, and the number of antecedents (number of nodes minus one) denotes their length. A given length often marks just a required minimum value, since the consequent 𝜑 itself may represent an implication, or even an implication chain, such hiding part of the whole chain. As an extension, it is useful to consider a single variable 𝜑 as a degenerate implication chain of length zero.

Implication chains play a particular role in logic, as all propositional expressions turn out to be convertible to one or more implication chains, their nodes as simple as a variable, or its negation.

So there is good reason to focus on implication chains as a sort of normalized expressions, and build some general theorems around them, with proofs using recursive patterns. This allows for theorems referring to longer and longer implication chains in an automated way.

The theorem names in this section contain the text fragment 'impchain' to point out their relevance to implication chains, followed by a number indicating the (minimal) length of the longest chain involved. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

𝜑       𝜑
 
Theoremwl-impchain-mp-x 34611 This series of theorems provide a means of exchanging the consequent of an implication chain via a simple implication. In the main part, the theorems ax-mp 5, syl 17, syl6 35, syl8 76 form the beginning of this series. These theorems are replicated here, but with proofs that aim at a recursive scheme, allowing to base a proof on that of the previous one in the series. (Contributed by Wolf Lammen, 17-Nov-2019.)
 
Theoremwl-impchain-mp-0 34612 This theorem is the start of a proof recursion scheme where we replace the consequent of an implication chain. The number '0' in the theorem name indicates that the modified chain has no antecedents.

This theorem is in fact a copy of ax-mp 5, and is repeated here to emphasize the recursion using similar theorem names. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

𝜓    &   (𝜓𝜑)       𝜑
 
Theoremwl-impchain-mp-1 34613 This theorem is in fact a copy of wl-luk-syl 34588, and repeated here to demonstrate a recursive proof scheme. The number '1' in the theorem name indicates that a chain of length 1 is modified. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜒𝜓)    &   (𝜓𝜑)       (𝜒𝜑)
 
Theoremwl-impchain-mp-2 34614 This theorem is in fact a copy of wl-luk-imtrdi 34596, and repeated here to demonstrate a recursive proof scheme. The number '2' in the theorem name indicates that a chain of length 2 is modified. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜃 → (𝜒𝜓))    &   (𝜓𝜑)       (𝜃 → (𝜒𝜑))
 
Theoremwl-impchain-com-1.x 34615 It is often convenient to have the antecedent under focus in first position, so we can apply immediate theorem forms (as opposed to deduction, tautology form). This series of theorems swaps the first with the last antecedent in an implication chain. This kind of swapping is self-inverse, whence we prefer it over, say, rotating theorems. A consequent can hide a tail of a longer chain, so theorems of this series appear as swapping a pair of antecedents with fixed offsets. This form of swapping antecedents is flexible enough to allow for any permutation of antecedents in an implication chain.

The first elements of this series correspond to com12 32, com13 88, com14 96 and com15 101 in the main part.

The proofs of this series aim at automated proving using a simple recursive scheme. It employs the previous theorem in the series along with a sample from the wl-impchain-mp-x 34611 series developed before. (Contributed by Wolf Lammen, 17-Nov-2019.)

 
Theoremwl-impchain-com-1.1 34616 A degenerate form of antecedent swapping. The number '1' in the theorem name indicates that it handles a chain of length 1.

Since there is just one antecedent in the chain, there is nothing to swap. Nondegenerated forms begin with wl-impchain-com-1.2 34617, for more see there. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

(𝜓𝜑)       (𝜓𝜑)
 
Theoremwl-impchain-com-1.2 34617 This theorem is in fact a copy of wl-luk-com12 34600, and repeated here to demonstrate a simple proof scheme. The number '2' in the theorem name indicates that a chain of length 2 is modified.

See wl-impchain-com-1.x 34615 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

(𝜒 → (𝜓𝜑))       (𝜓 → (𝜒𝜑))
 
Theoremwl-impchain-com-1.3 34618 This theorem is in fact a copy of com13 88, and repeated here to demonstrate a simple proof scheme. The number '3' in the theorem name indicates that a chain of length 3 is modified.

See wl-impchain-com-1.x 34615 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

(𝜃 → (𝜒 → (𝜓𝜑)))       (𝜓 → (𝜒 → (𝜃𝜑)))
 
Theoremwl-impchain-com-1.4 34619 This theorem is in fact a copy of com14 96, and repeated here to demonstrate a simple proof scheme. The number '4' in the theorem name indicates that a chain of length 4 is modified.

See wl-impchain-com-1.x 34615 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

(𝜂 → (𝜃 → (𝜒 → (𝜓𝜑))))       (𝜓 → (𝜃 → (𝜒 → (𝜂𝜑))))
 
Theoremwl-impchain-com-n.m 34620 This series of theorems allow swapping any two antecedents in an implication chain. The theorem names follow a pattern wl-impchain-com-n.m with integral numbers n < m, that swaps the m-th antecedent with n-th one in an implication chain. It is sufficient to restrict the length of the chain to m, too, since the consequent can be assumed to be the tail right of the m-th antecedent of any arbitrary sized implication chain. We further assume n > 1, since the wl-impchain-com-1.x 34615 series already covers the special case n = 1.

Being able to swap any two antecedents in an implication chain lays the foundation of permuting its antecedents arbitrarily.

The proofs of this series aim at automated proofing using a simple scheme. Any instance of this series is a triple step of swapping the first and n-th antecedent, then the first and the m-th, then the first and the n-th antecedent again. Each of these steps is an instance of the wl-impchain-com-1.x 34615 series. (Contributed by Wolf Lammen, 17-Nov-2019.)

 
Theoremwl-impchain-com-2.3 34621 This theorem is in fact a copy of com23 86. It starts a series of theorems named after wl-impchain-com-n.m 34620. For more information see there. (Contributed by Wolf Lammen, 12-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜃 → (𝜒 → (𝜓𝜑)))       (𝜃 → (𝜓 → (𝜒𝜑)))
 
Theoremwl-impchain-com-2.4 34622 This theorem is in fact a copy of com24 95. It is another instantiation of theorems named after wl-impchain-com-n.m 34620. For more information see there. (Contributed by Wolf Lammen, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜂 → (𝜃 → (𝜒 → (𝜓𝜑))))       (𝜂 → (𝜓 → (𝜒 → (𝜃𝜑))))
 
Theoremwl-impchain-com-3.2.1 34623 This theorem is in fact a copy of com3r 87. The proof is an example of how to arrive at arbitrary permutations of antecedents, using only swapping theorems. The recursion principle is to first swap the correct antecedent to the position just before the consequent, and then employ a theorem handling an implication chain of length one less to reorder the others. (Contributed by Wolf Lammen, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 → (𝜒 → (𝜓𝜑)))       (𝜓 → (𝜃 → (𝜒𝜑)))
 
Theoremwl-impchain-a1-x 34624 If an implication chain is assumed (hypothesis) or proven (theorem) to hold, then we may add any extra antecedent to it, without changing its truth. This is expressed in its simplest form in wl-luk-a1i 34593, that allows us prepending an arbitrary antecedent to an implication chain. Using our antecedent swapping theorems described in wl-impchain-com-n.m 34620, we may then move such a prepended antecedent to any desired location within all antecedents. The first series of theorems of this kind adds a single antecedent somewhere to an implication chain. The appended number in the theorem name indicates its position within all antecedents, 1 denoting the head position. A second theorem series extends this idea to multiple additions (TODO).

Adding antecedents to an implication chain usually weakens their universality. The consequent afterwards dependends on more conditions than before, which renders the implication chain less versatile. So you find this proof technique mostly when you adjust a chain to a hypothesis of a rule. A common case are syllogisms merging two implication chains into one.

The first elements of the first series correspond to a1i 11, a1d 25 and a1dd 50 in the main part.

The proofs of this series aim at automated proving using a simple recursive scheme. It employs the previous theorem in the series along with a sample from the wl-impchain-com-1.x 34615 series developed before. (Contributed by Wolf Lammen, 20-Jun-2020.)

 
Theoremwl-impchain-a1-1 34625 Inference rule, a copy of a1i 11. Head start of a recursive proof pattern. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑       (𝜓𝜑)
 
Theoremwl-impchain-a1-2 34626 Inference rule, a copy of a1d 25. First recursive proof based on the previous instance. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓))
 
Theoremwl-impchain-a1-3 34627 Inference rule, a copy of a1dd 50. A recursive proof depending on previous instances, and demonstrating the proof pattern. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜃𝜒)))
 
20.18.3  An alternative axiom ~ ax-13
 
Axiomax-wl-13v 34628* A version of ax13v 2384 with a distinctor instead of a distinct variable expression.

Had we additionally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1902. So essentially this theorem states, that a distinct variable condition between set variables can be replaced with a distinctor expression. (Contributed by Wolf Lammen, 23-Jul-2021.)

(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
 
Theoremwl-ax13lem1 34629* A version of ax-wl-13v 34628 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. This proof bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 23-Jul-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
 
20.18.4  Other stuff
 
Theoremwl-mps 34630 Replacing a nested consequent. A sort of modus ponens in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   ((𝜑𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)
 
Theoremwl-syls1 34631 Replacing a nested consequent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓𝜒)    &   ((𝜑𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)
 
Theoremwl-syls2 34632 Replacing a nested antecedent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   ((𝜑𝜒) → 𝜃)       ((𝜓𝜒) → 𝜃)
 
Theoremwl-embant 34633 A true wff can always be added as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremwl-orel12 34634 In a conjunctive normal form a pair of nodes like (𝜑𝜓) ∧ (¬ 𝜑𝜒) eliminates the need of a node (𝜓𝜒). This theorem allows simplifications in that respect. (Contributed by Wolf Lammen, 20-Jun-2020.)
(((𝜑𝜓) ∧ (¬ 𝜑𝜒)) → (𝜓𝜒))
 
Theoremwl-cases2-dnf 34635 A particular instance of orddi 1003 and anddi 1004 converting between disjunctive and conjunctive normal forms, when both 𝜑 and ¬ 𝜑 appear. This theorem in fact rephrases cases2 1039, and is related to consensus 1044. I restate it here in DNF and CNF. The proof deliberately does not use df-ifp 1055 and dfifp4 1058, by which it can be shortened. (Contributed by Wolf Lammen, 21-Jun-2020.) (Proof modification is discouraged.)
(((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))
 
Theoremwl-cbvmotv 34636* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by Wolf Lammen, 5-Mar-2023.)
(∃*𝑥⊤ → ∃*𝑦⊤)
 
Theoremwl-moteq 34637 Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by Wolf Lammen, 5-Mar-2023.)
(∃*𝑥⊤ → 𝑦 = 𝑧)
 
Theoremwl-motae 34638 Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by Wolf Lammen, 5-Mar-2023.)
(∃*𝑢⊤ → ∀𝑥 𝑦 = 𝑧)
 
Theoremwl-moae 34639* Two ways to express "at most one thing exists" or, in this context equivalently, "exactly one thing exists" . The equivalence results from the presence of ax-6 1961 in the proof, that ensures "at least one thing exists". For other equivalences see wl-euae 34640 and exists1 2744. Gerard Lang pointed out, that 𝑦𝑥𝑥 = 𝑦 with disjoint 𝑥 and 𝑦 (df-mo 2618, trut 1534) also means "exactly one thing exists" . (Contributed by NM, 5-Apr-2004.) State the theorem using truth constant . (Revised by BJ, 7-Oct-2022.) Reduce axiom dependencies, and use ∃*. (Revised by Wolf Lammen, 7-Mar-2023.)
(∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
 
Theoremwl-euae 34640* Two ways to express "exactly one thing exists" . (Contributed by Wolf Lammen, 5-Mar-2023.)
(∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
 
Theoremwl-nax6im 34641* The following series of theorems are centered around the empty domain, where no set exists. As a consequence, a set variable like 𝑥 has no instance to assign to. An expression like 𝑥 = 𝑦 is not really meaningful then. What does it evaluate to, true or false? In fact, the grammar extension weq 1955 requires us to formally assign a boolean value to an equation, say always false, unless you want to give up on exmid 888, for example. Whatever it is, we start out with the contraposition of ax-6 1961, that guarantees the existence of at least one set. Our hypothesis here expresses tentatively it might not hold. We can simplify the antecedent then, to the point where we do not need equation any more. This suggests what a decent characterization of the empty domain of discourse could be. (Contributed by Wolf Lammen, 12-Mar-2023.)
(¬ ∃𝑥 𝑥 = 𝑦𝜑)       (¬ ∃𝑥⊤ → 𝜑)
 
Theoremwl-hbae1 34642 This specialization of hbae 2448 does not depend on ax-11 2151. (Contributed by Wolf Lammen, 8-Aug-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦𝑥 𝑥 = 𝑦)
 
Theoremwl-naevhba1v 34643* An instance of hbn1w 2044 applied to equality. (Contributed by Wolf Lammen, 7-Apr-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)
 
Theoremwl-spae 34644 Prove an instance of sp 2172 from ax-13 2383 and Tarski's FOL only, without distinct variable conditions. The antecedent 𝑥𝑥 = 𝑦 holds in a multi-object universe only if 𝑦 is substituted for 𝑥, or vice versa, i.e. both variables are effectively the same. The converse ¬ ∀𝑥𝑥 = 𝑦 indicates that both variables are distinct, and it so provides a simple translation of a distinct variable condition to a logical term. In case studies 𝑥𝑥 = 𝑦 and ¬ ∀𝑥𝑥 = 𝑦 can help eliminating distinct variable conditions.

The antecedent 𝑥𝑥 = 𝑦 is expressed in the theorem's name by the abbreviation ae standing for 'all equal'.

Note that we cannot provide a logical predicate telling us directly whether a logical expression contains a particular variable, as such a construct would usually contradict ax-12 2167.

Note that this theorem is also provable from ax-12 2167 alone, so you can pick the axiom it is based on.

Compare this result to 19.3v 1977 and spaev 2048 having distinct variable conditions, but a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 5-Apr-2021.)

(∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
 
Theoremwl-speqv 34645* Under the assumption ¬ 𝑥 = 𝑦 a specialized version of sp 2172 is provable from Tarski's FOL and ax13v 2384 only. Note that this reverts the implication in ax13lem1 2385, so in fact 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
𝑥 = 𝑦 → (∀𝑥 𝑧 = 𝑦𝑧 = 𝑦))
 
Theoremwl-19.8eqv 34646* Under the assumption ¬ 𝑥 = 𝑦 a specialized version of 19.8a 2170 is provable from Tarski's FOL and ax13v 2384 only. Note that this reverts the implication in ax13lem2 2387, so in fact 𝑥 = 𝑦 → (∃𝑥𝑧 = 𝑦𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
𝑥 = 𝑦 → (𝑧 = 𝑦 → ∃𝑥 𝑧 = 𝑦))
 
Theoremwl-19.2reqv 34647* Under the assumption ¬ 𝑥 = 𝑦 the reverse direction of 19.2 1972 is provable from Tarski's FOL and ax13v 2384 only. Note that in conjunction with 19.2 1972 in fact 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦 ↔ ∃𝑥𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
 
Theoremwl-nfalv 34648* If 𝑥 is not present in 𝜑, it is not free in 𝑦𝜑. (Contributed by Wolf Lammen, 11-Jan-2020.)
𝑥𝑦𝜑
 
Theoremwl-nfimf1 34649 An antecedent is irrelevant to a not-free property, if it always holds. I used this variant of nfim 1888 in dvelimdf 2466 to simplify the proof. (Contributed by Wolf Lammen, 14-Oct-2018.)
(∀𝑥𝜑 → (Ⅎ𝑥(𝜑𝜓) ↔ Ⅎ𝑥𝜓))
 
Theoremwl-nfae1 34650 Unlike nfae 2450, this specialized theorem avoids ax-11 2151. (Contributed by Wolf Lammen, 26-Jun-2019.)
𝑥𝑦 𝑦 = 𝑥
 
Theoremwl-nfnae1 34651 Unlike nfnae 2451, this specialized theorem avoids ax-11 2151. (Contributed by Wolf Lammen, 27-Jun-2019.)
𝑥 ¬ ∀𝑦 𝑦 = 𝑥
 
Theoremwl-aetr 34652 A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧))
 
Theoremwl-dral1d 34653 A version of dral1 2456 with a context. Note: At first glance one might be tempted to generalize this (or a similar) theorem by weakening the first two hypotheses adding a 𝑥 = 𝑦, 𝑥𝑥 = 𝑦 or 𝜑 antecedent. wl-equsal1i 34665 and nf5di 2285 show that this is in fact pointless. (Contributed by Wolf Lammen, 28-Jul-2019.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)))
 
Theoremwl-cbvalnaed 34654 wl-cbvalnae 34655 with a context. (Contributed by Wolf Lammen, 28-Jul-2019.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜓))    &   (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 
Theoremwl-cbvalnae 34655 A more general version of cbval 2410 when non-free properties depend on a distinctor. Such expressions arise in proofs aiming at the elimination of distinct variable constraints, specifically in application of dvelimf 2465, nfsb2 2518 or dveeq1 2391. (Contributed by Wolf Lammen, 4-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)    &   (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)
 
Theoremwl-exeq 34656 The semantics of 𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.)
(∃𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))
 
Theoremwl-aleq 34657 The semantics of 𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.)
(∀𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∧ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧)))
 
Theoremwl-nfeqfb 34658 Extend nfeqf 2392 to an equivalence. (Contributed by Wolf Lammen, 31-Jul-2019.)
(Ⅎ𝑥 𝑦 = 𝑧 ↔ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑧))
 
Theoremwl-nfs1t 34659 If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Closed form of nfs1 2523. (Contributed by Wolf Lammen, 27-Jul-2019.)
(Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
 
Theoremwl-equsalvw 34660* Version of equsalv 2259 with a disjoint variable condition, and of equsal 2433 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsexvw 2002.

This theorem lays the foundation to a transformation of expressions called substitution of set variables in a wff. Only in this particular context we additionally assume 𝜑 and 𝑦 disjoint, stated here as 𝜑(𝑥). Similarly the disjointness of 𝜓 and 𝑥 is expressed by 𝜓(𝑦). Both 𝜑 and 𝜓 may still depend on other set variables, but that is irrelevant here.

We want to transform 𝜑(𝑥) into 𝜓(𝑦) such that 𝜓 depends on 𝑦 the same way as 𝜑 depends on 𝑥. This dependency is expressed in our hypothesis (called implicit substitution): (𝑥 = 𝑦 → (𝜑𝜓)). For primitive enough 𝜑 a sort of textual substitution of 𝑥 by 𝑦 is sufficient for such transformation. But note: 𝜑 must not contain wff variables, and the substitution is no proper textual substitution either. We still need grammar information to not accidently replace the x in a token 'x.' denoting multiplication, but only catch set variables 𝑥. Our current stage of development allows only equations and quantifiers make up such primitives. Thanks to equequ1 2023 and cbvalvw 2034 we can then prove in a mechanical way that in fact the implicit substitution holds for each instance.

If 𝜑 contains wff variables we cannot use textual transformation any longer, since we don't know how to replace 𝑦 for 𝑥 in placeholders of unknown structure. Our theorem now states, that the generic expression 𝑥(𝑥 = 𝑦𝜑) formally behaves as if such a substitution was possible and made.

(Contributed by BJ, 31-May-2019.)

(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 
Theoremwl-equsald 34661 Deduction version of equsal 2433. (Contributed by Wolf Lammen, 27-Jul-2019.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥(𝑥 = 𝑦𝜓) ↔ 𝜒))
 
Theoremwl-equsal 34662 A useful equivalence related to substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) It seems proving wl-equsald 34661 first, and then deriving more specialized versions wl-equsal 34662 and wl-equsal1t 34663 then is more efficient than the other way round, which is possible, too. See also equsal 2433. (Revised by Wolf Lammen, 27-Jul-2019.) (Proof modification is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 
Theoremwl-equsal1t 34663 The expression 𝑥 = 𝑦 in antecedent position plays an important role in predicate logic, namely in implicit substitution. However, occasionally it is irrelevant, and can safely be dropped. A sufficient condition for this is when 𝑥 (or 𝑦 or both) is not free in 𝜑.

This theorem is more fundamental than equsal 2433, spimt 2397 or sbft 2261, to which it is related. (Contributed by Wolf Lammen, 19-Aug-2018.)

(Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))
 
Theoremwl-equsalcom 34664 This simple equivalence eases substitution of one expression for the other. (Contributed by Wolf Lammen, 1-Sep-2018.)
(∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑦 = 𝑥𝜑))
 
Theoremwl-equsal1i 34665 The antecedent 𝑥 = 𝑦 is irrelevant, if one or both setvar variables are not free in 𝜑. (Contributed by Wolf Lammen, 1-Sep-2018.)
(Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑)    &   (𝑥 = 𝑦𝜑)       𝜑
 
Theoremwl-sb6rft 34666 A specialization of wl-equsal1t 34663. Closed form of sb6rf 2486. (Contributed by Wolf Lammen, 27-Jul-2019.)
(Ⅎ𝑥𝜑 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑)))
 
Theoremwl-cbvalsbi 34667* Change bounded variables in a special case. The reverse direction seems to involve ax-11 2151. My hope is that I will in some future be able to prove mo3 2644 with reversed quantifiers not using ax-11 2151. See also the remark in mo4 2646, which lead me to this effort. (Contributed by Wolf Lammen, 5-Mar-2024.)
(∀𝑥𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)
 
Theoremwl-sbrimt 34668 Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2305. (Contributed by Wolf Lammen, 26-Jul-2019.)
(Ⅎ𝑥𝜑 → ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)))
 
Theoremwl-sblimt 34669 Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2305. (Contributed by Wolf Lammen, 26-Jul-2019.)
(Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓)))
 
Theoremwl-sb8t 34670 Substitution of variable in universal quantifier. Closed form of sb8 2555. (Contributed by Wolf Lammen, 27-Jul-2019.)
(∀𝑥𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
 
Theoremwl-sb8et 34671 Substitution of variable in universal quantifier. Closed form of sb8e 2556. (Contributed by Wolf Lammen, 27-Jul-2019.)
(∀𝑥𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))
 
Theoremwl-sbhbt 34672 Closed form of sbhb 2559. Characterizing the expression 𝜑 → ∀𝑥𝜑 using a substitution expression. (Contributed by Wolf Lammen, 28-Jul-2019.)
(∀𝑥𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
 
Theoremwl-sbnf1 34673 Two ways expressing that 𝑥 is effectively not free in 𝜑. Simplified version of sbnf2 2370. Note: This theorem shows that sbnf2 2370 has unnecessary distinct variable constraints. (Contributed by Wolf Lammen, 28-Jul-2019.)
(∀𝑥𝑦𝜑 → (Ⅎ𝑥𝜑 ↔ ∀𝑥𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
 
Theoremwl-equsb3 34674 equsb3 2100 with a distinctor. (Contributed by Wolf Lammen, 27-Jun-2019.)
(¬ ∀𝑦 𝑦 = 𝑧 → ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧))
 
Theoremwl-equsb4 34675 Substitution applied to an atomic wff. The distinctor antecedent is more general than a distinct variable constraint. (Contributed by Wolf Lammen, 26-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑧 → ([𝑦 / 𝑥]𝑦 = 𝑧𝑦 = 𝑧))
 
Theoremwl-2sb6d 34676 Version of 2sb6 2085 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.)
(𝜑 → ¬ ∀𝑦 𝑦 = 𝑥)    &   (𝜑 → ¬ ∀𝑦 𝑦 = 𝑤)    &   (𝜑 → ¬ ∀𝑦 𝑦 = 𝑧)    &   (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧)       (𝜑 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜓 ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜓)))
 
Theoremwl-sbcom2d-lem1 34677* Lemma used to prove wl-sbcom2d 34679. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.)
((𝑢 = 𝑦𝑣 = 𝑤) → (¬ ∀𝑥 𝑥 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
 
Theoremwl-sbcom2d-lem2 34678* Lemma used to prove wl-sbcom2d 34679. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.)
(¬ ∀𝑦 𝑦 = 𝑥 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∀𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) → 𝜑)))
 
Theoremwl-sbcom2d 34679 Version of sbcom2 2158 with a context, and distinct variable conditions replaced with distinctors. (Contributed by Wolf Lammen, 4-Aug-2019.)
(𝜑 → ¬ ∀𝑥 𝑥 = 𝑤)    &   (𝜑 → ¬ ∀𝑥 𝑥 = 𝑧)    &   (𝜑 → ¬ ∀𝑧 𝑧 = 𝑦)       (𝜑 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜓))
 
Theoremwl-sbalnae 34680 A theorem used in elimination of disjoint variable restrictions by replacing them with distinctors. (Contributed by Wolf Lammen, 25-Jul-2019.)
((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 
Theoremwl-sbal1 34681* A theorem used in elimination of disjoint variable restriction on 𝑥 and 𝑦 by replacing it with a distinctor ¬ ∀𝑥𝑥 = 𝑧. (Contributed by NM, 15-May-1993.) Proof is based on wl-sbalnae 34680 now. See also sbal1 2568. (Revised by Wolf Lammen, 25-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 
Theoremwl-sbal2 34682* Move quantifier in and out of substitution. Revised to remove a distinct variable constraint. (Contributed by NM, 2-Jan-2002.) Proof is based on wl-sbalnae 34680 now. See also sbal2 2569. (Revised by Wolf Lammen, 25-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 
Theoremwl-2spsbbi 34683 spsbbi 2069 applied twice. (Contributed by Wolf Lammen, 5-Aug-2023.)
(∀𝑎𝑏(𝜑𝜓) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜓))
 
Theoremwl-lem-exsb 34684* This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)
(𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theoremwl-lem-nexmo 34685 This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)
(¬ ∃𝑥𝜑 → ∀𝑥(𝜑𝑥 = 𝑧))
 
Theoremwl-lem-moexsb 34686* The antecedent 𝑥(𝜑𝑥 = 𝑧) relates to ∃*𝑥𝜑, but is better suited for usage in proofs. Note that no distinct variable restriction is placed on 𝜑.

This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)

(∀𝑥(𝜑𝑥 = 𝑧) → (∃𝑥𝜑 ↔ [𝑧 / 𝑥]𝜑))
 
Theoremwl-alanbii 34687 This theorem extends alanimi 1808 to a biconditional. Recurrent usage stacks up more quantifiers. (Contributed by Wolf Lammen, 4-Oct-2019.)
(𝜑 ↔ (𝜓𝜒))       (∀𝑥𝜑 ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒))
 
Theoremwl-mo2df 34688 Version of mof 2643 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate disjoint variable conditions. (Contributed by Wolf Lammen, 11-Aug-2019.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)    &   (𝜑 → Ⅎ𝑦𝜓)       (𝜑 → (∃*𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦)))
 
Theoremwl-mo2tf 34689 Closed form of mof 2643 with a distinctor avoiding distinct variable conditions. (Contributed by Wolf Lammen, 20-Sep-2020.)
((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
 
Theoremwl-eudf 34690 Version of eu6 2655 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate disjoint variable conditions. (Contributed by Wolf Lammen, 23-Sep-2020.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)    &   (𝜑 → Ⅎ𝑦𝜓)       (𝜑 → (∃!𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦)))
 
Theoremwl-eutf 34691 Closed form of eu6 2655 with a distinctor avoiding distinct variable conditions. (Contributed by Wolf Lammen, 23-Sep-2020.)
((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑦𝜑) → (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
 
Theoremwl-euequf 34692 euequ 2679 proved with a distinctor. (Contributed by Wolf Lammen, 23-Sep-2020.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦)
 
Theoremwl-mo2t 34693* Closed form of mof 2643. (Contributed by Wolf Lammen, 18-Aug-2019.)
(∀𝑥𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
 
Theoremwl-mo3t 34694* Closed form of mo3 2644. (Contributed by Wolf Lammen, 18-Aug-2019.)
(∀𝑥𝑦𝜑 → (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
 
Theoremwl-sb8eut 34695 Substitution of variable in universal quantifier. Closed form of sb8eu 2682. (Contributed by Wolf Lammen, 11-Aug-2019.)
(∀𝑥𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))
 
Theoremwl-sb8mot 34696 Substitution of variable in universal quantifier. Closed form of sb8mo 2683.

This theorem relates to wl-mo3t 34694, since replacing 𝜑 with [𝑦 / 𝑥]𝜑 in the latter yields subexpressions like [𝑥 / 𝑦][𝑦 / 𝑥]𝜑, which can be reduced to 𝜑 via sbft 2261 and sbco 2545. So ∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑 is provable from wl-mo3t 34694 in a simple fashion, unfortunately only if 𝑥 and 𝑦 are known to be distinct. To avoid any hassle with distinctors, we prefer to derive this theorem independently, ignoring the close connection between both theorems. From an educational standpoint, one would assume wl-mo3t 34694 to be more fundamental, as it hints how the "at most one" objects on both sides of the biconditional correlate (they are the same), if they exist at all, and then prove this theorem from it. (Contributed by Wolf Lammen, 11-Aug-2019.)

(∀𝑥𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑))
 
Theoremwl-axc11rc11 34697 Proving axc11r 2379 from axc11 2447. The hypotheses are two instances of axc11 2447 used in the proof here. Some systems introduce axc11 2447 as an axiom, see for example System S2 in https://us.metamath.org/downloads/finiteaxiom.pdf .

By contrast, this database sees the variant axc11r 2379, directly derived from ax-12 2167, as foundational. Later axc11 2447 is proven somewhat trickily, requiring ax-10 2136 and ax-13 2383, see its proof. (Contributed by Wolf Lammen, 18-Jul-2023.)

(∀𝑦 𝑦 = 𝑥 → (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥))    &   (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))       (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
 
Axiomax-wl-11v 34698* Version of ax-11 2151 with distinct variable conditions. Currently implemented as an axiom to detect unintended references to the foundational axiom ax-11 2151. It will later be converted into a theorem directly based on ax-11 2151. (Contributed by Wolf Lammen, 28-Jun-2019.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
Theoremwl-ax11-lem1 34699 A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 ↔ ∀𝑦 𝑦 = 𝑧))
 
Theoremwl-ax11-lem2 34700* Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦)
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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 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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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