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Type | Label | Description | ||||||||||||||||||||||
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Statement | ||||||||||||||||||||||||
Theorem | wl-section-prop 35201 |
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 35202, ax-luk2 35203 and ax-luk3 35204. I rather copy this system than use luk-1 1662 to luk-3 1664, 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 | ||||||||||||||||||||||||
Axiom | ax-luk1 35202 |
1 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of
luk-1 1662 and imim1 83, 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 𝜑 → 𝜓 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 63 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 → (𝜑 → 𝜒))); (¬ 𝜑 → (𝜒 → (𝜑 → 𝜒))); (𝜓 → (¬ 𝜓 → (𝜑 → 𝜒))); (¬ 𝜑 → (¬ 𝜓 → (𝜑 → 𝜒))). (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) | ||||||||||||||||||||||||
Axiom | ax-luk2 35203 |
2 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of
luk-2 1663 or pm2.18 128, 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 pm2.65 196):
((𝜑 → 𝜑) → ((¬ 𝜑 → 𝜑) → 𝜑)). (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ ((¬ 𝜑 → 𝜑) → 𝜑) | ||||||||||||||||||||||||
Axiom | ax-luk3 35204 |
3 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of
luk-3 1664 and pm2.24 124, but introduced as an axiom.
One might think that the similar pm2.21 123 (¬ 𝜑 → (𝜑 → 𝜓)) 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 by
hand, that all possible interpretations of ax-mp 5, ax-luk1 35202, ax-luk2 35203
and pm2.21 123 result in phi1 or phi2, meaning they always hold. But for
wl-luk-ax3 35216 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 ----------------------------------------------------------------------
Implication is defined as ----------------------------------------------------------------------
(Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → (¬ 𝜑 → 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-section-boot 35205 | 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 | ||||||||||||||||||||||||
Theorem | wl-luk-imim1i 35206 | Inference adding common consequents in an implication, thereby interchanging the original antecedent and consequent. Copy of imim1i 63 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜓 → 𝜒) → (𝜑 → 𝜒)) | ||||||||||||||||||||||||
Theorem | wl-luk-syl 35207 | An inference version of the transitive laws for implication luk-1 1662. Copy of syl 17 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝜓) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) | ||||||||||||||||||||||||
Theorem | wl-luk-imtrid 35208 | A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. Copy of syl5 34 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → (𝜓 → 𝜃)) ⇒ ⊢ (𝜒 → (𝜑 → 𝜃)) | ||||||||||||||||||||||||
Theorem | wl-luk-pm2.18d 35209 | Deduction based on reductio ad absurdum. Copy of pm2.18d 127 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → (¬ 𝜓 → 𝜓)) ⇒ ⊢ (𝜑 → 𝜓) | ||||||||||||||||||||||||
Theorem | wl-luk-con4i 35210 | Inference rule. Copy of con4i 114 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (¬ 𝜑 → ¬ 𝜓) ⇒ ⊢ (𝜓 → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-luk-pm2.24i 35211 | Inference rule. Copy of pm2.24i 153 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ 𝜑 ⇒ ⊢ (¬ 𝜑 → 𝜓) | ||||||||||||||||||||||||
Theorem | wl-luk-a1i 35212 | 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 ⇒ ⊢ (𝜓 → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-luk-mpi 35213 | 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 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → 𝜒) | ||||||||||||||||||||||||
Theorem | wl-luk-imim2i 35214 | 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.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜒 → 𝜑) → (𝜒 → 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-luk-imtrdi 35215 | A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. Copy of syl6 35 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜒 → 𝜃) ⇒ ⊢ (𝜑 → (𝜓 → 𝜃)) | ||||||||||||||||||||||||
Theorem | wl-luk-ax3 35216 | 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) → (𝜓 → 𝜑)) | ||||||||||||||||||||||||
Theorem | wl-luk-ax1 35217 | ax-1 6 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → (𝜓 → 𝜑)) | ||||||||||||||||||||||||
Theorem | wl-luk-pm2.27 35218 | 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 42 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-luk-com12 35219 | 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 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜓 → (𝜑 → 𝜒)) | ||||||||||||||||||||||||
Theorem | wl-luk-pm2.21 35220 | 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 → (𝜑 → 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-luk-con1i 35221 | 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.) | ||||||||||||||||||||||
⊢ (¬ 𝜑 → 𝜓) ⇒ ⊢ (¬ 𝜓 → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-luk-ja 35222 | Inference joining the antecedents of two premises. Copy of ja 189 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (¬ 𝜑 → 𝜒) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ ((𝜑 → 𝜓) → 𝜒) | ||||||||||||||||||||||||
Theorem | wl-luk-imim2 35223 | 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.) | ||||||||||||||||||||||
⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓))) | ||||||||||||||||||||||||
Theorem | wl-luk-a1d 35224 | 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.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → (𝜒 → 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-luk-ax2 35225 | ax-2 7 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-luk-id 35226 | 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.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-luk-notnotr 35227 | 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.) | ||||||||||||||||||||||
⊢ (¬ ¬ 𝜑 → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-luk-pm2.04 35228 | 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 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-section-impchain 35229 |
An implication like (𝜓 → 𝜑) with one antecedent can easily be
extended by prepending more and more antecedents, as in
(𝜒
→ (𝜓 → 𝜑)) or (𝜃 → (𝜒 → (𝜓 → 𝜑))). 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 | ||||||||||||||||||||||||
Theorem | wl-impchain-mp-x 35230 | This series of theorems provide a means of exchanging the consequent of an implication chain via a simple implication. In the main part, 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.) | ||||||||||||||||||||||
⊢ ⊤ | ||||||||||||||||||||||||
Theorem | wl-impchain-mp-0 35231 |
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 & ⊢ (𝜓 → 𝜑) ⇒ ⊢ 𝜑 | ||||||||||||||||||||||||
Theorem | wl-impchain-mp-1 35232 | This theorem is in fact a copy of wl-luk-syl 35207, 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.) | ||||||||||||||||||||||
⊢ (𝜒 → 𝜓) & ⊢ (𝜓 → 𝜑) ⇒ ⊢ (𝜒 → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-impchain-mp-2 35233 | This theorem is in fact a copy of wl-luk-imtrdi 35215, 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 → (𝜒 → 𝜓)) & ⊢ (𝜓 → 𝜑) ⇒ ⊢ (𝜃 → (𝜒 → 𝜑)) | ||||||||||||||||||||||||
Theorem | wl-impchain-com-1.x 35234 |
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 35230 series developed before. (Contributed by Wolf Lammen, 17-Nov-2019.) | ||||||||||||||||||||||
⊢ ⊤ | ||||||||||||||||||||||||
Theorem | wl-impchain-com-1.1 35235 |
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 35236, for more see there. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜓 → 𝜑) ⇒ ⊢ (𝜓 → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-impchain-com-1.2 35236 |
This theorem is in fact a copy of wl-luk-com12 35219, 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 35234 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜒 → (𝜓 → 𝜑)) ⇒ ⊢ (𝜓 → (𝜒 → 𝜑)) | ||||||||||||||||||||||||
Theorem | wl-impchain-com-1.3 35237 |
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 35234 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 → (𝜓 → 𝜑))) ⇒ ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜑))) | ||||||||||||||||||||||||
Theorem | wl-impchain-com-1.4 35238 |
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 35234 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 → (𝜓 → 𝜑)))) ⇒ ⊢ (𝜓 → (𝜃 → (𝜒 → (𝜂 → 𝜑)))) | ||||||||||||||||||||||||
Theorem | wl-impchain-com-n.m 35239 |
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 35234 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 35234 series. (Contributed by Wolf Lammen, 17-Nov-2019.) | ||||||||||||||||||||||
⊢ ⊤ | ||||||||||||||||||||||||
Theorem | wl-impchain-com-2.3 35240 | This theorem is in fact a copy of com23 86. It starts a series of theorems named after wl-impchain-com-n.m 35239. For more information see there. (Contributed by Wolf Lammen, 12-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜃 → (𝜒 → (𝜓 → 𝜑))) ⇒ ⊢ (𝜃 → (𝜓 → (𝜒 → 𝜑))) | ||||||||||||||||||||||||
Theorem | wl-impchain-com-2.4 35241 | This theorem is in fact a copy of com24 95. It is another instantiation of theorems named after wl-impchain-com-n.m 35239. For more information see there. (Contributed by Wolf Lammen, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ (𝜂 → (𝜃 → (𝜒 → (𝜓 → 𝜑)))) ⇒ ⊢ (𝜂 → (𝜓 → (𝜒 → (𝜃 → 𝜑)))) | ||||||||||||||||||||||||
Theorem | wl-impchain-com-3.2.1 35242 | 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 → (𝜓 → 𝜑))) ⇒ ⊢ (𝜓 → (𝜃 → (𝜒 → 𝜑))) | ||||||||||||||||||||||||
Theorem | wl-impchain-a1-x 35243 |
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 35212, that
allows us prepending an arbitrary antecedent to an implication chain.
Using our antecedent swapping theorems described in
wl-impchain-com-n.m 35239, 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 35234 series developed before. (Contributed by Wolf Lammen, 20-Jun-2020.) | ||||||||||||||||||||||
⊢ ⊤ | ||||||||||||||||||||||||
Theorem | wl-impchain-a1-1 35244 | 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 ⇒ ⊢ (𝜓 → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-impchain-a1-2 35245 | 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.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → (𝜒 → 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-impchain-a1-3 35246 | 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 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-ifp-ncond1 35247 | If one case of an if- condition is false, the other automatically follows. (Contributed by Wolf Lammen, 21-Jul-2024.) | ||||||||||||||||||||||
⊢ (¬ 𝜓 → (if-(𝜑, 𝜓, 𝜒) ↔ (¬ 𝜑 ∧ 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-ifp-ncond2 35248 | If one case of an if- condition is false, the other automatically follows. (Contributed by Wolf Lammen, 21-Jul-2024.) | ||||||||||||||||||||||
⊢ (¬ 𝜒 → (if-(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∧ 𝜓))) | ||||||||||||||||||||||||
Theorem | wl-ifpimpr 35249 | If one case of an if- condition is a consequence of the other, the expression in df-ifp 1063 can be shortened. (Contributed by Wolf Lammen, 12-Jun-2024.) | ||||||||||||||||||||||
⊢ ((𝜒 → 𝜓) → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-ifp4impr 35250 | If one case of an if- condition is a consequence of the other, the expression in dfifp4 1066 can be shortened. (Contributed by Wolf Lammen, 18-Jun-2024.) | ||||||||||||||||||||||
⊢ ((𝜒 → 𝜓) → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ 𝜓))) | ||||||||||||||||||||||||
Theorem | wl-df-3xor 35251 |
Alternative definition of whad 1598 based on hadifp 1611. See df-had 1599 to
learn how it is currently introduced. The only use case so far is being a
binary addition primitive for df-sad 15887. If inputs are viewed as binary
digits (true is 1, false is 0), the result is what a binary single-bit
addition with carry-in yields in the low bit of their sum.
The core meaning is to check whether an odd number of three inputs are true. The ⊻ operation tests this for two inputs. So, if the first input is true, the two remaining inputs need to amount to an even (or: not an odd) number, else to an odd number. The idea of an odd number of inputs being true carries over to other than 3 inputs by recursion: In an informal notation we depend the case with n+1 inputs, 𝜑 being the additional one, recursively on that of n inputs: "(n+1)-xor" ↔ if-(𝜑, ¬ "n-xor" , "n-xor" ). The base case is "0-xor" being ⊥, because zero inputs never contain an odd number among them. Then we find, after simplifying, in our informal notation: "2-xor" (𝜑, 𝜓) ↔ (𝜑 ⊻ 𝜓) (see wl-2xor 35266). Our definition here follows exactly the above pattern. In microprocessor technology an addition limited to a range (a one-bit range in our case) is called a "wrap-around operation". The name "had", as in df-had 1599, by contrast, is somehow suggestive of a "half adder" instead. Such a circuit, for one, takes two inputs only, no carry-in, and then yields two outputs - both sum and carry. That's why we use "3xor" instead of "had" here. (Contributed by Wolf Lammen, 24-Apr-2024.) | ||||||||||||||||||||||
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ (𝜓 ⊻ 𝜒), (𝜓 ⊻ 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-df3xor2 35252 | Alternative definition of wl-df-3xor 35251, using triple exclusive disjunction, or XOR3. You can add more input by appending each one with a ⊻. Copy of hadass 1602. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 1-May-2024.) | ||||||||||||||||||||||
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓 ⊻ 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-df3xor3 35253 | Alternative form of wl-df3xor2 35252. Copy of df-had 1599. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 1-May-2024.) | ||||||||||||||||||||||
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ⊻ 𝜓) ⊻ 𝜒)) | ||||||||||||||||||||||||
Theorem | wl-3xortru 35254 | If the first input is true, then triple xor is equivalent to the biconditionality of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 24-Apr-2024.) | ||||||||||||||||||||||
⊢ (𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ ¬ (𝜓 ⊻ 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-3xorfal 35255 | If the first input is false, then triple xor is equivalent to the exclusive disjunction of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 29-Apr-2024.) | ||||||||||||||||||||||
⊢ (¬ 𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ⊻ 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-3xorbi 35256 | Triple xor can be replaced with a triple biconditional. Unlike ⊻, you cannot add more inputs by simply stacking up more biconditionals, and still express an "odd number of inputs". (Contributed by Wolf Lammen, 24-Apr-2024.) | ||||||||||||||||||||||
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-3xorbi2 35257 | Alternative form of wl-3xorbi 35256. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 24-Apr-2024.) | ||||||||||||||||||||||
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ 𝜒)) | ||||||||||||||||||||||||
Theorem | wl-3xorbi123d 35258 | Equivalence theorem for triple xor. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 24-Apr-2024.) | ||||||||||||||||||||||
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → (𝜃 ↔ 𝜏)) & ⊢ (𝜑 → (𝜂 ↔ 𝜁)) ⇒ ⊢ (𝜑 → (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁))) | ||||||||||||||||||||||||
Theorem | wl-3xorbi123i 35259 | Equivalence theorem for triple xor. Copy of hadbi123i 1601. (Contributed by Mario Carneiro, 4-Sep-2016.) | ||||||||||||||||||||||
⊢ (𝜓 ↔ 𝜒) & ⊢ (𝜃 ↔ 𝜏) & ⊢ (𝜂 ↔ 𝜁) ⇒ ⊢ (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁)) | ||||||||||||||||||||||||
Theorem | wl-3xorrot 35260 | Rotation law for triple xor. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 24-Apr-2024.) | ||||||||||||||||||||||
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜒, 𝜑)) | ||||||||||||||||||||||||
Theorem | wl-3xorcoma 35261 | Commutative law for triple xor. Copy of hadcoma 1604. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 17-Dec-2023.) | ||||||||||||||||||||||
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜑, 𝜒)) | ||||||||||||||||||||||||
Theorem | wl-3xorcomb 35262 | Commutative law for triple xor. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 24-Apr-2024.) | ||||||||||||||||||||||
⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜑, 𝜒, 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-3xornot1 35263 | Flipping the first input flips the triple xor. wl-3xorrot 35260 can rotate any input to the front, so flipping any one of them does the same. (Contributed by Wolf Lammen, 1-May-2024.) | ||||||||||||||||||||||
⊢ (¬ hadd(𝜑, 𝜓, 𝜒) ↔ hadd(¬ 𝜑, 𝜓, 𝜒)) | ||||||||||||||||||||||||
Theorem | wl-3xornot 35264 | Triple xor distributes over negation. Copy of hadnot 1608. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.) | ||||||||||||||||||||||
⊢ (¬ hadd(𝜑, 𝜓, 𝜒) ↔ hadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒)) | ||||||||||||||||||||||||
Theorem | wl-1xor 35265 |
In the recursive scheme
"(n+1)-xor" ↔ if-(𝜑, ¬ "n-xor" , "n-xor" ) we set n = 0 to formally arrive at an expression for "1-xor". The base case "0-xor" is replaced with ⊥, as a sequence of 0 inputs never has an odd number being part of it. (Contributed by Wolf Lammen, 11-May-2024.) | ||||||||||||||||||||||
⊢ (if-(𝜓, ¬ ⊥, ⊥) ↔ 𝜓) | ||||||||||||||||||||||||
Theorem | wl-2xor 35266 |
In the recursive scheme
"(n+1)-xor" ↔ if-(𝜑, ¬ "n-xor" , "n-xor" ) we set n = 1 to formally arrive at an expression for "2-xor". It is based on "1-xor", that is known to be equivalent to its only input (see wl-1xor 35265). (Contributed by Wolf Lammen, 11-May-2024.) | ||||||||||||||||||||||
⊢ (if-(𝜑, ¬ 𝜓, 𝜓) ↔ (𝜑 ⊻ 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-df-3mintru2 35267 |
Alternative definition of wcad 1612. See df-cad 1613 to learn how it is
currently introduced. The only use case so far is being a binary addition
primitive for df-sad 15887. If inputs are viewed as binary digits
(true is
1, false is 0), the result is whether ordinary binary full addition yields
a carry bit. That is what the name df-cad 1613 is derived from: "carry of
an addition". Here we stick with this abbreviated form of our
notation
above, but still use "adder carry" as a shorthand for "at
least 2 out of
3" in text.
The core meaning is to check whether at least two of three inputs are true. So, if the first input is true, at least one of the two remaining must be true, else even both. This theorem is the in-between of "at least 1 out of 3", given by triple disjunction df-3or 1089, and "(at least) 3 out of 3", expressed by triple conjunction df-3an 1090. The notion above can be generalized to other input numbers with other minimum values as follows. Let us introduce informally a logical operation "n-mintru-m" taking n inputs, and requiring at least m of them be true to let the operation itself be true. There now exists a recursive scheme to define it for increasing n, m. We start with the base case n = 0. Here "n-mintru-0" is equivalent to ⊤ (any sequence of inputs contains at least zero true inputs), the other "0-mintru-m" is for any m > 0 equivalent to ⊥, because a sequence of zero inputs never has a positive number of them true. The general case adds a new input 𝜑 to a given sequence of n inputs, and reduces that case for all integers m to that of the smaller sequence by recursion, informally written as: "(n+1)-mintru-(m+1)" ↔ if-(𝜑, "n-mintru-m" , "n-mintru-(m+1)" ) Our definition here matches "3-mintru-2" with inputs 𝜑, 𝜓 and 𝜒. Starting from the base cases we find after simplifications: "2-mintru-2" (𝜓, 𝜒) ↔ (𝜓 ∧ 𝜒) (wl-2mintru2 35274), and "2-mintru-1" (𝜓, 𝜒) ↔ (𝜓 ∨ 𝜒) (wl-2mintru1 35273). Plugging these expressions into the formula above for n = 3, m = 2 yields exactly our definition here. (Contributed by Wolf Lammen, 2-May-2024.) | ||||||||||||||||||||||
⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 ∨ 𝜒), (𝜓 ∧ 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-df2-3mintru2 35268 | The adder carry in disjunctive normal form. An alternative highly symmetric definition emphasizing the independence of order of the inputs 𝜑, 𝜓 and 𝜒. Copy of cador 1614. (Contributed by Mario Carneiro, 4-Sep-2016.) df-cad redefined. (Revised by Wolf Lammen, 12-Jun-2024.) | ||||||||||||||||||||||
⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-df3-3mintru2 35269 | The adder carry in conjunctive normal form. An alternative highly symmetric definition emphasizing the independence of order of the inputs 𝜑, 𝜓 and 𝜒. Copy of cadan 1615. (Contributed by Mario Carneiro, 4-Sep-2016.) df-cad redefined. (Revised by Wolf Lammen, 18-Jun-2024.) | ||||||||||||||||||||||
⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-df4-3mintru2 35270 | An alternative definition of the adder carry. Copy of df-cad 1613. (Contributed by Mario Carneiro, 4-Sep-2016.) df-cad redefined. (Revised by Wolf Lammen, 19-Jun-2024.) | ||||||||||||||||||||||
⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓)))) | ||||||||||||||||||||||||
Theorem | wl-1mintru1 35271 |
Using the recursion formula:
"(n+1)-mintru-(m+1)" ↔ if-(𝜑, "n-mintru-m" , "n-mintru-(m+1)" ) for "1-mintru-1" (meaning "at least 1 out of 1 input is true") by plugging in n = 0, m = 0, and simplifying. The expressions "0-mintru-0" and "0-mintru-1" are base cases of the recursion, meaning "in a sequence of zero inputs, at least 0 / 1 input is true", respectively equvalent to ⊤ / ⊥. Negating an "n-mintru1" operation means: All n inputs 𝜑.. 𝜃 are false. This is also conveniently expressed as ¬ (𝜑 ∨.. ∨ 𝜃). Applying this idea here (n = 1) yields the obvious result that in an input sequence of size 1 only then all will be false, if its single input is. (Contributed by Wolf Lammen, 10-May-2024.) | ||||||||||||||||||||||
⊢ (if-(𝜒, ⊤, ⊥) ↔ 𝜒) | ||||||||||||||||||||||||
Theorem | wl-1mintru2 35272 |
Using the recursion formula:
"(n+1)-mintru-(m+1)" ↔ if-(𝜑, "n-mintru-m" , "n-mintru-(m+1)" ) for "1-mintru-2" (meaning "at least 2 out of a single input are true") by plugging in n = 0, m = 1, and simplifying. The expressions "0-mintru-1" and "0-mintru-2" are base cases of the recursion, meaning "in a sequence of zero inputs at least 1 / 2 input is true", evaluate both to ⊥. Since no sequence of inputs has a longer subsequence of whatever property, the resulting ⊥ is to be expected. Negating a "n-mintru2" operation has an interesting interpretation: at most one input is true, so all inputs exclude each other mutually. Such an exclusion is expressed by a NAND operation (𝜑 ⊼ 𝜓), not by a XOR. Applying this idea here (n = 1) leads to the obvious "In a single input sequence 'at most one is true' always holds". (Contributed by Wolf Lammen, 10-May-2024.) | ||||||||||||||||||||||
⊢ (if-(𝜒, ⊥, ⊥) ↔ ⊥) | ||||||||||||||||||||||||
Theorem | wl-2mintru1 35273 |
Using the recursion formula
"(n+1)-mintru-(m+1)" ↔ if-(𝜑, "n-mintru-m" , "n-mintru-(m+1)" ) for "2-mintru-1" (meaning "at least 1 out of 2 inputs is true") by plugging in n = 1, m = 0, and simplifying. The expression "1-mintru-0" is a base case (meaning at least zero inputs out of 1 are true), evaluating to ⊤, and wl-1mintru1 35271 shows "1-mintru-1" is equivalent to the only input. Negating an "n-mintru1" operation means: All n inputs 𝜑.. 𝜃 are false. This is also conveniently expressed as ¬ (𝜑 ∨.. ∨ 𝜃), in accordance with the result here. (Contributed by Wolf Lammen, 10-May-2024.) | ||||||||||||||||||||||
⊢ (if-(𝜓, ⊤, 𝜒) ↔ (𝜓 ∨ 𝜒)) | ||||||||||||||||||||||||
Theorem | wl-2mintru2 35274 |
Using the recursion formula
"(n+1)-mintru-(m+1)" ↔ if-(𝜑, "n-mintru-m" , "n-mintru-(m+1)" ) for "2-mintru-2" (meaning "2 out of 2 inputs are true") by plugging in n = 1, m = 1, and simplifying. See wl-1mintru1 35271 and wl-1mintru2 35272 to see that "1-mintru-1" / "1-mintru-2" evaluate to 𝜒 / ⊥ respectively. Negating a "n-mintru2" operation means 'at most one input is true', so all inputs exclude each other mutually. Such an exclusion is expressed by a NAND operation (𝜑 ⊼ 𝜓), not by a XOR. Applying this idea here (n = 2) yields the expected NAND in case of a pair of inputs. (Contributed by Wolf Lammen, 10-May-2024.) | ||||||||||||||||||||||
⊢ (if-(𝜓, 𝜒, ⊥) ↔ (𝜓 ∧ 𝜒)) | ||||||||||||||||||||||||
Theorem | wl-df3maxtru1 35275 |
Assuming "(n+1)-maxtru1" ↔ ¬
"(n+1)-mintru-2", we can deduce from
the recursion formula given in wl-df-3mintru2 35267, that a similiar one
"(n+1)-maxtru1" ↔ if-(𝜑,-. "n-mintru-1" , "n-maxtru1" ) is valid for expressing 'at most one input is true'. This can also be rephrased as a mutual exclusivity of propositional expressions (no two of a sequence of inputs can simultaniously be true). Of course, this suggests that all inputs depend on variables 𝜂, 𝜁... Whatever wellformed expression we plugin for these variables, it will render at most one of the inputs true. The here introduced mutual exclusivity is possibly useful for case studies, where we want the cases be sort of 'disjoint'. One can further imagine that a complete case scenario demands that the 'at most' is sharpened to 'exactly one'. This does not impose any difficulty here, as one of the inputs will then be the negation of all others be or'ed. As one input is determined, 'at most one' is sufficient to describe the general form here. Since cadd is an alias for 'at least 2 out of three are true', its negation is under focus here. (Contributed by Wolf Lammen, 23-Jun-2024.) | ||||||||||||||||||||||
⊢ (¬ cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 ⊽ 𝜒), (𝜓 ⊼ 𝜒))) | ||||||||||||||||||||||||
Axiom | ax-wl-13v 35276* |
A version of ax13v 2372 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 1916. 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)) | ||||||||||||||||||||||||
Theorem | wl-ax13lem1 35277* | A version of ax-wl-13v 35276 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)) | ||||||||||||||||||||||||
Theorem | wl-mps 35278 | 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 → (𝜓 → 𝜒)) & ⊢ ((𝜑 → 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 → 𝜓) → 𝜃) | ||||||||||||||||||||||||
Theorem | wl-syls1 35279 | 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.) | ||||||||||||||||||||||
⊢ (𝜓 → 𝜒) & ⊢ ((𝜑 → 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 → 𝜓) → 𝜃) | ||||||||||||||||||||||||
Theorem | wl-syls2 35280 | 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.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝜓) & ⊢ ((𝜑 → 𝜒) → 𝜃) ⇒ ⊢ ((𝜓 → 𝜒) → 𝜃) | ||||||||||||||||||||||||
Theorem | wl-embant 35281 | 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 & ⊢ (𝜓 → 𝜒) ⇒ ⊢ ((𝜑 → 𝜓) → 𝜒) | ||||||||||||||||||||||||
Theorem | wl-orel12 35282 | In a conjunctive normal form a pair of nodes like (𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒) eliminates the need of a node (𝜓 ∨ 𝜒). This theorem allows simplifications in that respect. (Contributed by Wolf Lammen, 20-Jun-2020.) | ||||||||||||||||||||||
⊢ (((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒)) → (𝜓 ∨ 𝜒)) | ||||||||||||||||||||||||
Theorem | wl-cases2-dnf 35283 | A particular instance of orddi 1009 and anddi 1010 converting between disjunctive and conjunctive normal forms, when both 𝜑 and ¬ 𝜑 appear. This theorem in fact rephrases cases2 1047, and is related to consensus 1052. I restate it here in DNF and CNF. The proof deliberately does not use df-ifp 1063 and dfifp4 1066, by which it can be shortened. (Contributed by Wolf Lammen, 21-Jun-2020.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-cbvmotv 35284* | 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⊤) | ||||||||||||||||||||||||
Theorem | wl-moteq 35285 | 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) | ||||||||||||||||||||||||
Theorem | wl-motae 35286 | 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) | ||||||||||||||||||||||||
Theorem | wl-moae 35287* | 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 1974 in the proof, that ensures "at least one thing exists". For other equivalences see wl-euae 35288 and exists1 2663. Gerard Lang pointed out, that ∃𝑦∀𝑥𝑥 = 𝑦 with disjoint 𝑥 and 𝑦 (df-mo 2540, trut 1548) 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) | ||||||||||||||||||||||||
Theorem | wl-euae 35288* | Two ways to express "exactly one thing exists" . (Contributed by Wolf Lammen, 5-Mar-2023.) | ||||||||||||||||||||||
⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦) | ||||||||||||||||||||||||
Theorem | wl-nax6im 35289* | 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 1969 requires us to formally assign a boolean value to an equation, say always false, unless you want to give up on exmid 894, for example. Whatever it is, we start out with the contraposition of ax-6 1974, 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 = 𝑦 → 𝜑) ⇒ ⊢ (¬ ∃𝑥⊤ → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-hbae1 35290 | This specialization of hbae 2430 does not depend on ax-11 2161. (Contributed by Wolf Lammen, 8-Aug-2021.) | ||||||||||||||||||||||
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦) | ||||||||||||||||||||||||
Theorem | wl-naevhba1v 35291* | An instance of hbn1w 2057 applied to equality. (Contributed by Wolf Lammen, 7-Apr-2021.) | ||||||||||||||||||||||
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦) | ||||||||||||||||||||||||
Theorem | wl-spae 35292 |
Prove an instance of sp 2183 from ax-13 2371 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 2178. Note that this theorem is also provable from ax-12 2178 alone, so you can pick the axiom it is based on. Compare this result to 19.3v 1990 and spaev 2061 having distinct variable conditions, but a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 5-Apr-2021.) | ||||||||||||||||||||||
⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦) | ||||||||||||||||||||||||
Theorem | wl-speqv 35293* | Under the assumption ¬ 𝑥 = 𝑦 a specialized version of sp 2183 is provable from Tarski's FOL and ax13v 2372 only. Note that this reverts the implication in ax13lem1 2373, so in fact (¬ 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦 ↔ 𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.) | ||||||||||||||||||||||
⊢ (¬ 𝑥 = 𝑦 → (∀𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦)) | ||||||||||||||||||||||||
Theorem | wl-19.8eqv 35294* | Under the assumption ¬ 𝑥 = 𝑦 a specialized version of 19.8a 2181 is provable from Tarski's FOL and ax13v 2372 only. Note that this reverts the implication in ax13lem2 2375, so in fact (¬ 𝑥 = 𝑦 → (∃𝑥𝑧 = 𝑦 ↔ 𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.) | ||||||||||||||||||||||
⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∃𝑥 𝑧 = 𝑦)) | ||||||||||||||||||||||||
Theorem | wl-19.2reqv 35295* | Under the assumption ¬ 𝑥 = 𝑦 the reverse direction of 19.2 1985 is provable from Tarski's FOL and ax13v 2372 only. Note that in conjunction with 19.2 1985 in fact (¬ 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦 ↔ ∃𝑥𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.) | ||||||||||||||||||||||
⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | ||||||||||||||||||||||||
Theorem | wl-nfalv 35296* | If 𝑥 is not present in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by Wolf Lammen, 11-Jan-2020.) | ||||||||||||||||||||||
⊢ Ⅎ𝑥∀𝑦𝜑 | ||||||||||||||||||||||||
Theorem | wl-nfimf1 35297 | An antecedent is irrelevant to a not-free property, if it always holds. I used this variant of nfim 1902 in dvelimdf 2448 to simplify the proof. (Contributed by Wolf Lammen, 14-Oct-2018.) | ||||||||||||||||||||||
⊢ (∀𝑥𝜑 → (Ⅎ𝑥(𝜑 → 𝜓) ↔ Ⅎ𝑥𝜓)) | ||||||||||||||||||||||||
Theorem | wl-nfae1 35298 | Unlike nfae 2432, this specialized theorem avoids ax-11 2161. (Contributed by Wolf Lammen, 26-Jun-2019.) | ||||||||||||||||||||||
⊢ Ⅎ𝑥∀𝑦 𝑦 = 𝑥 | ||||||||||||||||||||||||
Theorem | wl-nfnae1 35299 | Unlike nfnae 2433, this specialized theorem avoids ax-11 2161. (Contributed by Wolf Lammen, 27-Jun-2019.) | ||||||||||||||||||||||
⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑥 | ||||||||||||||||||||||||
Theorem | wl-aetr 35300 | A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.) | ||||||||||||||||||||||
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧)) |
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