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Theorem List for Metamath Proof Explorer - 32401-32500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcsbfinxpg 32401* Distribute proper substitution through Cartesian exponentiation. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥(𝑈↑↑𝑁) = (𝐴 / 𝑥𝑈↑↑𝐴 / 𝑥𝑁))

Theoremfinxpreclem1 32402* Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.)
(𝑋𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑋⟩))

Theoremfinxpreclem2 32403* Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.)
((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ¬ ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑋⟩))

Theoremfinxp0 32404 The value of Cartesian exponentiation at zero. (Contributed by ML, 24-Oct-2020.)
(𝑈↑↑∅) = ∅

Theoremfinxp1o 32405 The value of Cartesian exponentiation at one. (Contributed by ML, 17-Oct-2020.)
(𝑈↑↑1𝑜) = 𝑈

Theoremfinxpreclem3 32406* Lemma for ↑↑ recursion theorems. (Contributed by ML, 20-Oct-2020.)
𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))       (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))

Theoremfinxpreclem4 32407* Lemma for ↑↑ recursion theorems. (Contributed by ML, 23-Oct-2020.)
𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))       (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁))

Theoremfinxpreclem5 32408* Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.)
𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))       ((𝑛 ∈ ω ∧ 1𝑜𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩))

Theoremfinxpreclem6 32409* Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.)
𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))       ((𝑁 ∈ ω ∧ 1𝑜𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))

Theoremfinxpsuclem 32410* Lemma for finxpsuc 32411. (Contributed by ML, 24-Oct-2020.)
𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))       ((𝑁 ∈ ω ∧ 1𝑜𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))

Theoremfinxpsuc 32411 The value of Cartesian exponentiation at a successor. (Contributed by ML, 24-Oct-2020.)
((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))

Theoremfinxp2o 32412 The value of Cartesian exponentiation at two. (Contributed by ML, 19-Oct-2020.)
(𝑈↑↑2𝑜) = (𝑈 × 𝑈)

Theoremfinxp3o 32413 The value of Cartesian exponentiation at three. (Contributed by ML, 24-Oct-2020.)
(𝑈↑↑3𝑜) = ((𝑈 × 𝑈) × 𝑈)

Theoremfinxpnom 32414 Cartesian exponentiation when the exponent is not a natural number defaults to the empty set. (Contributed by ML, 24-Oct-2020.)
𝑁 ∈ ω → (𝑈↑↑𝑁) = ∅)

Theoremfinxp00 32415 Cartesian exponentiation of the empty set to any power is the empty set. (Contributed by ML, 24-Oct-2020.)
(∅↑↑𝑁) = ∅

21.17  Mathbox for Wolf Lammen

Theoremwl-section-prop 32416 Intuitionistic logic is now developed separately, so we need not first focus on intuitionally valid axioms ax-1 6 and ax-2 7 any longer.

Alternatively, I start from Jan Lukasiewicz's axiom system here, i.e. ax-mp 5, ax-luk1 32417, ax-luk2 32418 and ax-luk3 32419. I rather copy this system than use luk-1 1571 to luk-3 1573, since the latter are theorems, while we need axioms here.

(Contributed by Wolf Lammen, 23-Feb-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

𝜑       𝜑

Axiomax-luk1 32417 1 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-1 1571 and imim1 81, but introduced as an axiom. It focuses on a basic property of a valid implication, namely that the consequent has to be true whenever the antecedent is. So if 𝜑 and 𝜓 are somehow parametrized expressions, then 𝜑𝜓 states that 𝜑 strengthen 𝜓, in that 𝜑 holds only for a (often proper) subset of those parameters making 𝜓 true. We easily accept, that when 𝜓 is stronger than 𝜒 and, at the same time 𝜑 is stronger than 𝜓, then 𝜑 must be stronger than 𝜒. This transitivity is expressed in this axiom.

A particular result of this strengthening property comes into play if the antecedent holds unconditionally. Then the consequent must hold unconditionally as well. This specialization is the foundational idea behind logical conclusion. Such conclusion is best expressed in so-called immediate versions of this axiom like imim1i 61 or syl 17. Note that these forms are weaker replacements (i.e. just frequent specialization) of the closed form presented here, hence a mere convenience.

We can identify in this axiom up to three antecedents, followed by a consequent. The number of antecedents is not really fixed; the fewer we are willing to "see", the more complex the consequent grows. On the other side, since 𝜒 is a variable capable of assuming an implication itself, we might find even more antecedents after some substitution of 𝜒. This shows that the ideas of antecedent and consequent in expressions like this depends on, and can adapt to, our current interpretation of the whole expression.

In this axiom, up to two antecedents happen to be of complex nature themselves, i.e. are an embedded implication. Logically, this axiom is a compact notion of simpler expressions, which I will later coin implication chains. Herein all antecedents and the consequent appear as simple variables, or their negation. Any propositional expression is equivalent to a set of such chains. This axiom, for example, is dissected into following chains, from which it can be recovered losslessly:

(𝜓 → (𝜒 → (𝜑𝜒))); 𝜑 → (𝜒 → (𝜑𝜒))); (𝜓 → (¬ 𝜓 → (𝜑𝜒))); 𝜑 → (¬ 𝜓 → (𝜑𝜒))). (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)

((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Axiomax-luk2 32418 2 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-2 1572 or pm2.18 121, but introduced as an axiom. The core idea behind this axiom is, that if something can be implied from both an antecedent, and separately from its negation, then the antecedent is irrelevant to the consequent, and can safely be dropped. This is perhaps better seen from the following slightly extended version (related to pm2.65 183):

((𝜑𝜑) → ((¬ 𝜑𝜑) → 𝜑)). (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)

((¬ 𝜑𝜑) → 𝜑)

Axiomax-luk3 32419 3 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-3 1573 and pm2.24 120, but introduced as an axiom. One might think that the similar pm2.21 119 𝜑 → (𝜑𝜓)) is a valid replacement for this axiom. But this is not true, ax-3 8 is not derivable from this modification. This can be shown by designing carefully operators ¬ and on a finite set of primitive statements. In propositional logic such statements are and , but we can assume more and other primitives in our universe of statements. So we denote our primitive statements as phi0 , phi1 and phi2. The actual meaning of the statements are not important in this context, it rather counts how they behave under our operations ¬ and , and which of them we assume to hold unconditionally (phi1, phi2). For our disproving model, I give that information in tabular form below. The interested reader may check per hand, that all possible interpretations of ax-mp 5, ax-luk1 32417, ax-luk2 32418 and pm2.21 119 result in phi1 or phi2, meaning they always hold. But for wl-ax3 32431 we can find a counter example resulting in phi0, not a statement always true. The verification of a particular set of axioms in a given model is tedious and error prone, so I wrote a computer program, first checking this for me, and second, hunting for a counter example. Here is the result, after 9165 fruitlessly computer generated models:

ax-3 fails for phi2, phi2
number of statements: 3
always true phi1 phi2

Negation is defined as
----------------------------------------------------------------------
 -. phi0 -. phi1 -. phi2 phi1 phi0 phi1

Implication is defined as
----------------------------------------------------------------------
 p->q q: phi0 q: phi1 q: phi2 p: phi0 phi1 phi1 phi1 p: phi1 phi0 phi1 phi1 p: phi2 phi0 phi0 phi0

(Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)
(𝜑 → (¬ 𝜑𝜓))

21.17.1  1. Bootstrapping

Theoremwl-section-boot 32420 In this section, I provide the first steps needed for convenient proving. The presented theorems follow no common concept other than being useful in themselves, and apt to rederive ax-1 6, ax-2 7 and ax-3 8. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑       𝜑

Theoremwl-imim1i 32421 Inference adding common consequents in an implication, thereby interchanging the original antecedent and consequent. Copy of imim1i 61 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.)
(𝜑𝜓)       ((𝜓𝜒) → (𝜑𝜒))

Theoremwl-syl 32422 An inference version of the transitive laws for implication luk-1 1571. Copy of syl 17 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)

Theoremwl-syl5 32423 A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. Copy of syl5 33 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜃))

Theoremwl-pm2.18d 32424 Deduction based on reductio ad absurdum. Copy of pm2.18d 123 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (¬ 𝜓𝜓))       (𝜑𝜓)

Theoremwl-con4i 32425 Inference rule. Copy of con4i 112 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑 → ¬ 𝜓)       (𝜓𝜑)

Theoremwl-pm2.24i 32426 Inference rule. Copy of pm2.24i 145 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑       𝜑𝜓)

Theoremwl-a1i 32427 Inference rule. Copy of a1i 11 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑       (𝜓𝜑)

Theoremwl-mpi 32428 A nested modus ponens inference. Copy of mpi 20 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜓    &   (𝜑 → (𝜓𝜒))       (𝜑𝜒)

Theoremwl-imim2i 32429 Inference adding common antecedents in an implication. Copy of imim2i 16 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜓)       ((𝜒𝜑) → (𝜒𝜓))

Theoremwl-syl6 32430 A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. Copy of syl6 34 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑 → (𝜓𝜃))

Theoremwl-ax3 32431 ax-3 8 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑))

Theoremwl-ax1 32432 ax-1 6 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜑))

Theoremwl-pm2.27 32433 This theorem, called "Assertion," can be thought of as closed form of modus ponens ax-mp 5. Theorem *2.27 of [WhiteheadRussell] p. 104. Copy of pm2.27 41 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → ((𝜑𝜓) → 𝜓))

Theoremwl-com12 32434 Inference that swaps (commutes) antecedents in an implication. Copy of com12 32 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜓 → (𝜑𝜒))

Theoremwl-pm2.21 32435 From a wff and its negation, anything follows. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. Copy of pm2.21 119 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑 → (𝜑𝜓))

Theoremwl-con1i 32436 A contraposition inference. Copy of con1i 143 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑𝜓)       𝜓𝜑)

Theoremwl-ja 32437 Inference joining the antecedents of two premises. Copy of ja 172 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑𝜒)    &   (𝜓𝜒)       ((𝜑𝜓) → 𝜒)

Theoremwl-imim2 32438 A closed form of syllogism (see syl 17). Theorem *2.05 of [WhiteheadRussell] p. 100. Copy of imim2 56 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))

Theoremwl-a1d 32439 Deduction introducing an embedded antecedent. Copy of imim2 56 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓))

Theoremwl-ax2 32440 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-id 32441 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-notnotr 32442 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 124 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(¬ ¬ 𝜑𝜑)

Theoremwl-pm2.04 32443 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 88 with a different proof. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))

21.17.2  Implication chains

Theoremwl-section-impchain 32444 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 32445 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 34, syl8 74 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 32446 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 32447 This theorem is in fact a copy of wl-syl 32422, 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 32448 This theorem is in fact a copy of wl-syl6 32430, 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 32449 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 86, com14 94 and com15 99 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 32445 series developed before. (Contributed by Wolf Lammen, 17-Nov-2019.)

Theoremwl-impchain-com-1.1 32450 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. Non-degenerated forms begin with wl-impchain-com-1.2 32451, 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 32451 This theorem is in fact a copy of wl-com12 32434, 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 32449 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 32452 This theorem is in fact a copy of com13 86, 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 32449 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 32453 This theorem is in fact a copy of com14 94, 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 32449 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 32454 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 32449 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 32449 series. (Contributed by Wolf Lammen, 17-Nov-2019.)

Theoremwl-impchain-com-2.3 32455 This theorem is in fact a copy of com23 84. It starts a series of theorems named after wl-impchain-com-n.m 32454. 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 32456 This theorem is in fact a copy of com24 93. It is another instantiation of theorems named after wl-impchain-com-n.m 32454. 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 32457 This theorem is in fact a copy of com3r 85. 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 32458 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-a1i 32427, that allows us prepending an arbitrary antecedent to an implication chain. Using our antecedent swapping theorems described in wl-impchain-com-n.m 32454, 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 48 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 32449 series developed before. (Contributed by Wolf Lammen, 20-Jun-2020.)

Theoremwl-impchain-a1-1 32459 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 32460 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 32461 Inference rule, a copy of a1dd 48. 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)))

21.17.3  An alternative definition of df-nf

Theoremwl-section-nf 32462 The current definition of 'not free', nf5 2102, has its downsides. In particular, it often drags axioms ax-10 2006 and ax-12 2034 into proofs, that are not needed otherwise. This is because of the particular structure of the term 𝑥(𝜑 → ∀𝑥𝜑). It does not allow an easy transition 𝜑--> ¬ 𝜑 (see nfn 1768). The mix of both quantified and simple 𝜑 requires explicit use of sp 2041 (or ax-12 2034) in many instances. And, finally, the nesting of the quantifier 𝑥 sometimes requires an invocation of theorems like nfa1 2015. All of this mandates the use of ax-10 2006 and/or ax-12 2034.

On the other hand, the current definition is structurally better aligned with both the hb* series of theorems and ax-5 1827.

A note on ax-10 2006: The obvious content of this axiom is, that the ¬ operator does not change the not-free state of a set varable. This is in fact only one aspect of this axiom. The other one, more hidden, states that 𝑥 is not free in 𝑥𝜑. A simple transformation renders this axiom as (∃𝑥𝑥𝜑 → ∀𝑥𝜑), from which the second aspect is better deduced. Both aspects are not needed in simple applications of the df-nf 1701 style definition, while use of the current nf5 2102 incurs these.

A note on ax-12 2034: This axiom enters proofs using via sp 2041 or 19.8a 2039 or their variants. In a context where both mixed quantified and simple variables 𝜑 appear (like 19.21 2062), this axiom is almost always required, no matter how 'not free' is defined. But in a context, where a variable 𝜑 appears only quantified, chances are, this axiom can be evaded when using df-nf 1701, but not when using nf5 2102.

(Contributed by Wolf Lammen, 11-Sep-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

𝜑       𝜑

Theoremwl-nf-nf2 32463 By ax-10 2006 the definition nf5 2102 is stricter than the df-nf 1701 style. (Contributed by Wolf Lammen, 14-Sep-2021.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)       (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑))

Theoremwl-nf2-nf 32464 hba1 2137 is sufficient to let a df-nf 1701 style definition be stricter than nf5 2102. (Contributed by Wolf Lammen, 14-Sep-2021.)
(∀𝑥𝜑 → ∀𝑥𝑥𝜑)       ((∃𝑥𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑))

21.17.4  An alternative axiom ~ ax-13

Axiomax-wl-13v 32465* A version of ax13v 2235 with a distinctor instead of a distinct variable expression.

Had we additonally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1827. 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 32466* A version of ax-wl-13v 32465 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))

21.17.5  Other stuff

Theoremwl-jarri 32467 Dropping a nested antecedent. This theorem is one of two reversions of ja 172. Since ja 172 is reversible, a nested (chain of) implication(s) is just a packed notation of two or more theorems/hypotheses with a common consequent. axc5c7 33214 is an instance of this idea. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)       (𝜓𝜒)

Theoremwl-jarli 32468 Dropping a nested consequent. This theorem is one of two reversions of ja 172. Since ja 172 is reversible, one can conclude, that a nested (chain of) implication(s) is just a packed notation of two or more theorems/ hypotheses with a common consequent. axc5c7 33214 is an instance of this idea. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)       𝜑𝜒)

Theoremwl-mps 32469 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 32470 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 32471 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 32472 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 32473 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 32474 A particular instance of orddi 909 and anddi 910 converting between disjunctive and conjunctive normal forms, when both 𝜑 and ¬ 𝜑 appear. This theorem in fact rephrases cases2 1005, and is related to consensus 990. I restate it here in DNF and CNF. The proof deliberately does not use df-ifp 1007 and dfifp4 1010, by which it can be shortened. (Contributed by Wolf Lammen, 21-Jun-2020.) (Proof modification is discouraged.)
(((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))

Theoremwl-dfnan2 32475 An alternative definition of "nand" based on imnan 437. See df-nan 1440 for the original definition. This theorem allows various shortenings. (Contributed by Wolf Lammen, 26-Jun-2020.)
((𝜑𝜓) ↔ (𝜑 → ¬ 𝜓))

Theoremwl-nancom 32476 The 'nand' operator commutes. (Contributed by Mario Carneiro, 9-May-2015.) (Revised by Wolf Lammen, 26-Jun-2020.)
((𝜑𝜓) ↔ (𝜓𝜑))

Theoremwl-nannan 32477 Lemma for handling nested 'nand's. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Revised by Wolf Lammen, 26-Jun-2020.)
((𝜑 ⊼ (𝜓𝜒)) ↔ (𝜑 → (𝜓𝜒)))

Theoremwl-nannot 32478 Show equivalence between negation and the Nicod version. To derive nic-dfneg 1586, apply nanbi 1446. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Revised by Wolf Lammen, 26-Jun-2020.)
𝜑 ↔ (𝜑𝜑))

Theoremwl-nanbi1 32479 Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.) (Revised by Wolf Lammen, 27-Jun-2020.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theoremwl-nanbi2 32480 Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.) (Revised by Wolf Lammen, 27-Jun-2020.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theoremwl-naev 32481* 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)

Theoremwl-hbae1 32482 This specialization of hbae 2303 does not depend on ax-11 2021. (Contributed by Wolf Lammen, 8-Aug-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦𝑥 𝑥 = 𝑦)

Theoremwl-naevhba1v 32483* An instance of hbn1w 1960 applied to equality. (Contributed by Wolf Lammen, 7-Apr-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)

Theoremwl-hbnaev 32484* Any variable is free in ¬ ∀𝑥𝑥 = 𝑦, if 𝑥 and 𝑦 are distinct. The latter condition can actually be lifted, but this version is easier to prove. The proof does not use ax-10 2006. (Contributed by Wolf Lammen, 9-Apr-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Theoremwl-spae 32485 Prove an instance of sp 2041 from ax-13 2234 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 2034.

Note that this theorem is also provable from ax-12 2034 alone, so you can pick the axiom it is based on.

Compare this result to 19.3v 1884 and spaev 1965 having distinct variable conditions, but a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 5-Apr-2021.)

(∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)

Theoremwl-cbv3vv 32486* Avoiding ax-11 2021. (Contributed by Wolf Lammen, 30-Aug-2021.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremwl-speqv 32487* Under the assumption ¬ 𝑥 = 𝑦 a specialized version of sp 2041 is provable from Tarski's FOL and ax13v 2235 only. Note that this reverts the implication in ax13lem1 2236, 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 32488* Under the assumption ¬ 𝑥 = 𝑦 a specialized version of 19.8a 2039 is provable from Tarski's FOL and ax13v 2235 only. Note that this reverts the implication in ax13lem2 2284, 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 32489* Under the assumption ¬ 𝑥 = 𝑦 the reverse direction of 19.2 1879 is provable from Tarski's FOL and ax13v 2235 only. Note that in conjunction with 19.2 1879 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-dveeq12 32490* The current form of ax-13 2234 has a particular disadvantage: The condition ¬ 𝑥 = 𝑦 is less versatile than the general form ¬ ∀𝑥𝑥 = 𝑦. You need ax-10 2006 to arrive at the more general form presented here. You need 19.8a 2039 (or ax-12 2034) to restore 𝑦 = 𝑧 from 𝑥𝑦 = 𝑧 again. (Contributed by Wolf Lammen, 9-Jun-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremwl-nfalv 32491* If 𝑥 is not present in 𝜑, it is not free in 𝑦𝜑. (Contributed by Wolf Lammen, 11-Jan-2020.)
𝑥𝑦𝜑

Theoremwl-nfimf1 32492 An antecedent is irrelevant to a not-free property, if it always holds. I used this variant of nfim 1813 in dvelimdf 2323 to simplify the proof. (Contributed by Wolf Lammen, 14-Oct-2018.)
(∀𝑥𝜑 → (Ⅎ𝑥(𝜑𝜓) ↔ Ⅎ𝑥𝜓))

Theoremwl-nfnbi 32493 Being free does not depend on an outside negation in an expression. This theorem is slightly more general than nfn 1768 or nfnd 1769. (Contributed by Wolf Lammen, 5-May-2018.)
(Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)

Theoremwl-nfae1 32494 Unlike nfae 2304, this specialized theorem avoids ax-11 2021. (Contributed by Wolf Lammen, 26-Jun-2019.)
𝑥𝑦 𝑦 = 𝑥

Theoremwl-nfnae1 32495 Unlike nfnae 2306, this specialized theorem avoids ax-11 2021. (Contributed by Wolf Lammen, 27-Jun-2019.)
𝑥 ¬ ∀𝑦 𝑦 = 𝑥

Theoremwl-aetr 32496 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 32497 A version of dral1 2313 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 32508 and nf5di 2105 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 32498 wl-cbvalnae 32499 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 32499 A more general version of cbval 2259 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 2322, nfsb2 2348 or dveeq1 2288. (Contributed by Wolf Lammen, 4-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)    &   (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremwl-exeq 32500 The semantics of 𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.)
(∃𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 42301-42360
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