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Theorem List for Metamath Proof Explorer - 36601-36700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempibt1 36601* Theorem T000001 of pi-base. A compact topology is also countably compact. See pibp16 36598 and pibp19 36599 for the definitions of the relevant properties. (Contributed by ML, 8-Dec-2020.)
𝐢 = {π‘₯ ∈ Top ∣ βˆ€π‘¦ ∈ 𝒫 π‘₯((βˆͺ π‘₯ = βˆͺ 𝑦 ∧ 𝑦 β‰Ό Ο‰) β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)βˆͺ π‘₯ = βˆͺ 𝑧)}    β‡’   (𝐽 ∈ Comp β†’ 𝐽 ∈ 𝐢)
 
Theorempibt2 36602* Theorem T000002 of pi-base, a countably compact topology is also weakly countably compact. See pibp19 36599 and pibp21 36600 for the definitions of the relevant properties. This proof uses the axiom of choice. (Contributed by ML, 30-Mar-2021.)
𝑋 = βˆͺ 𝐽    &   πΆ = {π‘₯ ∈ Top ∣ βˆ€π‘¦ ∈ 𝒫 π‘₯((βˆͺ π‘₯ = βˆͺ 𝑦 ∧ 𝑦 β‰Ό Ο‰) β†’ βˆƒπ‘§ ∈ (𝒫 𝑦 ∩ Fin)βˆͺ π‘₯ = βˆͺ 𝑧)}    &   π‘Š = {π‘₯ ∈ Top ∣ βˆ€π‘¦ ∈ (𝒫 βˆͺ π‘₯ βˆ– Fin)βˆƒπ‘§ ∈ βˆͺ π‘₯𝑧 ∈ ((limPtβ€˜π‘₯)β€˜π‘¦)}    β‡’   (𝐽 ∈ 𝐢 β†’ 𝐽 ∈ π‘Š)
 
21.20  Mathbox for Wolf Lammen
 
Theoremwl-section-prop 36603 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 36604, ax-luk2 36605 and ax-luk3 36606. I rather copy this system than use luk-1 1656 to luk-3 1658, 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.)

πœ‘    β‡’   πœ‘
 
Axiomax-luk1 36604 1 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-1 1656 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 πœ‘ and πœ“ are somehow parametrized expressions, then πœ‘ β†’ πœ“ states that πœ‘ strengthen πœ“, in that πœ‘ holds only for a (often proper) subset of those parameters making πœ“ true. We easily accept, that when πœ“ is stronger than πœ’ and, at the same time πœ‘ is stronger than πœ“, then πœ‘ must be stronger than πœ’. 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 πœ’ is a variable capable of assuming an implication itself, we might find even more antecedents after some substitution of πœ’. 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:

(πœ“ β†’ (πœ’ β†’ (πœ‘ β†’ πœ’))); (Β¬ πœ‘ β†’ (πœ’ β†’ (πœ‘ β†’ πœ’))); (πœ“ β†’ (Β¬ πœ“ β†’ (πœ‘ β†’ πœ’))); (Β¬ πœ‘ β†’ (Β¬ πœ“ β†’ (πœ‘ β†’ πœ’))). (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)

((πœ‘ β†’ πœ“) β†’ ((πœ“ β†’ πœ’) β†’ (πœ‘ β†’ πœ’)))
 
Axiomax-luk2 36605 2 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-2 1657 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 pm2.65 192):

((πœ‘ β†’ πœ‘) β†’ ((Β¬ πœ‘ β†’ πœ‘) β†’ πœ‘)). (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)

((Β¬ πœ‘ β†’ πœ‘) β†’ πœ‘)
 
Axiomax-luk3 36606 3 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-3 1658 and pm2.24 124, but introduced as an axiom. One might think that the similar pm2.21 123 (Β¬ πœ‘ β†’ (πœ‘ β†’ πœ“)) 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 Β¬ and β†’ on a finite set of primitive statements. In propositional logic such statements are ⊀ and βŠ₯, 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 Β¬ and β†’, 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 36604, ax-luk2 36605 and pm2.21 123 result in phi1 or phi2, meaning they always hold. But for wl-luk-ax3 36618 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
phi1phi0phi1

Implication is defined as
----------------------------------------------------------------------
p->qq: phi0q: phi1q: phi2
p: phi0phi1phi1phi1
p: phi1phi0phi1phi1
p: phi2phi0phi0phi0

(Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)
(πœ‘ β†’ (Β¬ πœ‘ β†’ πœ“))
 
21.20.1  1. Bootstrapping
 
Theoremwl-section-boot 36607 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.)
πœ‘    β‡’   πœ‘
 
Theoremwl-luk-imim1i 36608 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.)
(πœ‘ β†’ πœ“)    β‡’   ((πœ“ β†’ πœ’) β†’ (πœ‘ β†’ πœ’))
 
Theoremwl-luk-syl 36609 An inference version of the transitive laws for implication luk-1 1656. Copy of syl 17 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(πœ‘ β†’ πœ“)    &   (πœ“ β†’ πœ’)    β‡’   (πœ‘ β†’ πœ’)
 
Theoremwl-luk-imtrid 36610 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.)
(πœ‘ β†’ πœ“)    &   (πœ’ β†’ (πœ“ β†’ πœƒ))    β‡’   (πœ’ β†’ (πœ‘ β†’ πœƒ))
 
Theoremwl-luk-pm2.18d 36611 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.)
(πœ‘ β†’ (Β¬ πœ“ β†’ πœ“))    β‡’   (πœ‘ β†’ πœ“)
 
Theoremwl-luk-con4i 36612 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.)
(Β¬ πœ‘ β†’ Β¬ πœ“)    β‡’   (πœ“ β†’ πœ‘)
 
Theoremwl-luk-pm2.24i 36613 Inference rule. Copy of pm2.24i 150 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
πœ‘    β‡’   (Β¬ πœ‘ β†’ πœ“)
 
Theoremwl-luk-a1i 36614 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.)
πœ‘    β‡’   (πœ“ β†’ πœ‘)
 
Theoremwl-luk-mpi 36615 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.)
πœ“    &   (πœ‘ β†’ (πœ“ β†’ πœ’))    β‡’   (πœ‘ β†’ πœ’)
 
Theoremwl-luk-imim2i 36616 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.)
(πœ‘ β†’ πœ“)    β‡’   ((πœ’ β†’ πœ‘) β†’ (πœ’ β†’ πœ“))
 
Theoremwl-luk-imtrdi 36617 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.)
(πœ‘ β†’ (πœ“ β†’ πœ’))    &   (πœ’ β†’ πœƒ)    β‡’   (πœ‘ β†’ (πœ“ β†’ πœƒ))
 
Theoremwl-luk-ax3 36618 ax-3 8 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((Β¬ πœ‘ β†’ Β¬ πœ“) β†’ (πœ“ β†’ πœ‘))
 
Theoremwl-luk-ax1 36619 ax-1 6 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(πœ‘ β†’ (πœ“ β†’ πœ‘))
 
Theoremwl-luk-pm2.27 36620 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.)
(πœ‘ β†’ ((πœ‘ β†’ πœ“) β†’ πœ“))
 
Theoremwl-luk-com12 36621 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.)
(πœ‘ β†’ (πœ“ β†’ πœ’))    β‡’   (πœ“ β†’ (πœ‘ β†’ πœ’))
 
Theoremwl-luk-pm2.21 36622 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.)
(Β¬ πœ‘ β†’ (πœ‘ β†’ πœ“))
 
Theoremwl-luk-con1i 36623 A contraposition inference. Copy of con1i 147 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(Β¬ πœ‘ β†’ πœ“)    β‡’   (Β¬ πœ“ β†’ πœ‘)
 
Theoremwl-luk-ja 36624 Inference joining the antecedents of two premises. Copy of ja 186 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(Β¬ πœ‘ β†’ πœ’)    &   (πœ“ β†’ πœ’)    β‡’   ((πœ‘ β†’ πœ“) β†’ πœ’)
 
Theoremwl-luk-imim2 36625 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.)
((πœ‘ β†’ πœ“) β†’ ((πœ’ β†’ πœ‘) β†’ (πœ’ β†’ πœ“)))
 
Theoremwl-luk-a1d 36626 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.)
(πœ‘ β†’ πœ“)    β‡’   (πœ‘ β†’ (πœ’ β†’ πœ“))
 
Theoremwl-luk-ax2 36627 ax-2 7 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((πœ‘ β†’ (πœ“ β†’ πœ’)) β†’ ((πœ‘ β†’ πœ“) β†’ (πœ‘ β†’ πœ’)))
 
Theoremwl-luk-id 36628 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.)
(πœ‘ β†’ πœ‘)
 
Theoremwl-luk-notnotr 36629 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 πœ‘, then πœ‘ is stable. Copy of notnotr 130 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(Β¬ Β¬ πœ‘ β†’ πœ‘)
 
Theoremwl-luk-pm2.04 36630 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.)
((πœ‘ β†’ (πœ“ β†’ πœ’)) β†’ (πœ“ β†’ (πœ‘ β†’ πœ’)))
 
21.20.2  Implication chains
 
Theoremwl-section-impchain 36631 An implication like (πœ“ β†’ πœ‘) with one antecedent can easily be extended by prepending more and more antecedents, as in (πœ’ β†’ (πœ“ β†’ πœ‘)) or (πœƒ β†’ (πœ’ β†’ (πœ“ β†’ πœ‘))). 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 πœ‘ 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 πœ‘ 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.)

πœ‘    β‡’   πœ‘
 
Theoremwl-impchain-mp-x 36632 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.)
⊀
 
Theoremwl-impchain-mp-0 36633 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.)

πœ“    &   (πœ“ β†’ πœ‘)    β‡’   πœ‘
 
Theoremwl-impchain-mp-1 36634 This theorem is in fact a copy of wl-luk-syl 36609, 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.)
(πœ’ β†’ πœ“)    &   (πœ“ β†’ πœ‘)    β‡’   (πœ’ β†’ πœ‘)
 
Theoremwl-impchain-mp-2 36635 This theorem is in fact a copy of wl-luk-imtrdi 36617, 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.)
(πœƒ β†’ (πœ’ β†’ πœ“))    &   (πœ“ β†’ πœ‘)    β‡’   (πœƒ β†’ (πœ’ β†’ πœ‘))
 
Theoremwl-impchain-com-1.x 36636 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 36632 series developed before. (Contributed by Wolf Lammen, 17-Nov-2019.)

⊀
 
Theoremwl-impchain-com-1.1 36637 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 36638, for more see there. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

(πœ“ β†’ πœ‘)    β‡’   (πœ“ β†’ πœ‘)
 
Theoremwl-impchain-com-1.2 36638 This theorem is in fact a copy of wl-luk-com12 36621, 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 36636 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

(πœ’ β†’ (πœ“ β†’ πœ‘))    β‡’   (πœ“ β†’ (πœ’ β†’ πœ‘))
 
Theoremwl-impchain-com-1.3 36639 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 36636 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

(πœƒ β†’ (πœ’ β†’ (πœ“ β†’ πœ‘)))    β‡’   (πœ“ β†’ (πœ’ β†’ (πœƒ β†’ πœ‘)))
 
Theoremwl-impchain-com-1.4 36640 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 36636 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

(πœ‚ β†’ (πœƒ β†’ (πœ’ β†’ (πœ“ β†’ πœ‘))))    β‡’   (πœ“ β†’ (πœƒ β†’ (πœ’ β†’ (πœ‚ β†’ πœ‘))))
 
Theoremwl-impchain-com-n.m 36641 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 36636 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 36636 series. (Contributed by Wolf Lammen, 17-Nov-2019.)

⊀
 
Theoremwl-impchain-com-2.3 36642 This theorem is in fact a copy of com23 86. It starts a series of theorems named after wl-impchain-com-n.m 36641. For more information see there. (Contributed by Wolf Lammen, 12-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(πœƒ β†’ (πœ’ β†’ (πœ“ β†’ πœ‘)))    β‡’   (πœƒ β†’ (πœ“ β†’ (πœ’ β†’ πœ‘)))
 
Theoremwl-impchain-com-2.4 36643 This theorem is in fact a copy of com24 95. It is another instantiation of theorems named after wl-impchain-com-n.m 36641. For more information see there. (Contributed by Wolf Lammen, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(πœ‚ β†’ (πœƒ β†’ (πœ’ β†’ (πœ“ β†’ πœ‘))))    β‡’   (πœ‚ β†’ (πœ“ β†’ (πœ’ β†’ (πœƒ β†’ πœ‘))))
 
Theoremwl-impchain-com-3.2.1 36644 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.)
(πœƒ β†’ (πœ’ β†’ (πœ“ β†’ πœ‘)))    β‡’   (πœ“ β†’ (πœƒ β†’ (πœ’ β†’ πœ‘)))
 
Theoremwl-impchain-a1-x 36645 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 36614, that allows prepending an arbitrary antecedent to an implication chain. Using our antecedent swapping theorems described in wl-impchain-com-n.m 36641, 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 36636 series developed before. (Contributed by Wolf Lammen, 20-Jun-2020.)

⊀
 
Theoremwl-impchain-a1-1 36646 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.)
πœ‘    β‡’   (πœ“ β†’ πœ‘)
 
Theoremwl-impchain-a1-2 36647 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.)
(πœ‘ β†’ πœ“)    β‡’   (πœ‘ β†’ (πœ’ β†’ πœ“))
 
Theoremwl-impchain-a1-3 36648 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.)
(πœ‘ β†’ (πœ“ β†’ πœ’))    β‡’   (πœ‘ β†’ (πœ“ β†’ (πœƒ β†’ πœ’)))
 
21.20.3  Theorems around the conditional operator
 
Theoremwl-ifp-ncond1 36649 If one case of an if- condition is false, the other automatically follows. (Contributed by Wolf Lammen, 21-Jul-2024.)
(Β¬ πœ“ β†’ (if-(πœ‘, πœ“, πœ’) ↔ (Β¬ πœ‘ ∧ πœ’)))
 
Theoremwl-ifp-ncond2 36650 If one case of an if- condition is false, the other automatically follows. (Contributed by Wolf Lammen, 21-Jul-2024.)
(Β¬ πœ’ β†’ (if-(πœ‘, πœ“, πœ’) ↔ (πœ‘ ∧ πœ“)))
 
Theoremwl-ifpimpr 36651 If one case of an if- condition is a consequence of the other, the expression in df-ifp 1061 can be shortened. (Contributed by Wolf Lammen, 12-Jun-2024.)
((πœ’ β†’ πœ“) β†’ (if-(πœ‘, πœ“, πœ’) ↔ ((πœ‘ ∧ πœ“) ∨ πœ’)))
 
Theoremwl-ifp4impr 36652 If one case of an if- condition is a consequence of the other, the expression in dfifp4 1064 can be shortened. (Contributed by Wolf Lammen, 18-Jun-2024.)
((πœ’ β†’ πœ“) β†’ (if-(πœ‘, πœ“, πœ’) ↔ ((πœ‘ ∨ πœ’) ∧ πœ“)))
 
21.20.4  Alternative development of hadd, cadd
 
Theoremwl-df-3xor 36653 Alternative definition of whad 1593 based on hadifp 1605. See df-had 1594 to learn how it is currently introduced. The only use case so far is being a binary addition primitive for df-sad 16397. 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, πœ‘ being the additional one, recursively on that of n inputs: "(n+1)-xor" ↔ if-(πœ‘, Β¬ "n-xor" , "n-xor" ). The base case is "0-xor" being βŠ₯, because zero inputs never contain an odd number among them. Then we find, after simplifying, in our informal notation:

"2-xor" (πœ‘, πœ“) ↔ (πœ‘ ⊻ πœ“) (see wl-2xor 36668).

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 1594, 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(πœ‘, πœ“, πœ’) ↔ if-(πœ‘, Β¬ (πœ“ ⊻ πœ’), (πœ“ ⊻ πœ’)))
 
Theoremwl-df3xor2 36654 Alternative definition of wl-df-3xor 36653, using triple exclusive disjunction, or XOR3. You can add more input by appending each one with a ⊻. Copy of hadass 1597. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 1-May-2024.)
(hadd(πœ‘, πœ“, πœ’) ↔ (πœ‘ ⊻ (πœ“ ⊻ πœ’)))
 
Theoremwl-df3xor3 36655 Alternative form of wl-df3xor2 36654. Copy of df-had 1594. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 1-May-2024.)
(hadd(πœ‘, πœ“, πœ’) ↔ ((πœ‘ ⊻ πœ“) ⊻ πœ’))
 
Theoremwl-3xortru 36656 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.)
(πœ‘ β†’ (hadd(πœ‘, πœ“, πœ’) ↔ Β¬ (πœ“ ⊻ πœ’)))
 
Theoremwl-3xorfal 36657 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.)
(Β¬ πœ‘ β†’ (hadd(πœ‘, πœ“, πœ’) ↔ (πœ“ ⊻ πœ’)))
 
Theoremwl-3xorbi 36658 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(πœ‘, πœ“, πœ’) ↔ (πœ‘ ↔ (πœ“ ↔ πœ’)))
 
Theoremwl-3xorbi2 36659 Alternative form of wl-3xorbi 36658. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 24-Apr-2024.)
(hadd(πœ‘, πœ“, πœ’) ↔ ((πœ‘ ↔ πœ“) ↔ πœ’))
 
Theoremwl-3xorbi123d 36660 Equivalence theorem for triple xor. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 24-Apr-2024.)
(πœ‘ β†’ (πœ“ ↔ πœ’))    &   (πœ‘ β†’ (πœƒ ↔ 𝜏))    &   (πœ‘ β†’ (πœ‚ ↔ 𝜁))    β‡’   (πœ‘ β†’ (hadd(πœ“, πœƒ, πœ‚) ↔ hadd(πœ’, 𝜏, 𝜁)))
 
Theoremwl-3xorbi123i 36661 Equivalence theorem for triple xor. Copy of hadbi123i 1596. (Contributed by Mario Carneiro, 4-Sep-2016.)
(πœ“ ↔ πœ’)    &   (πœƒ ↔ 𝜏)    &   (πœ‚ ↔ 𝜁)    β‡’   (hadd(πœ“, πœƒ, πœ‚) ↔ hadd(πœ’, 𝜏, 𝜁))
 
Theoremwl-3xorrot 36662 Rotation law for triple xor. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 24-Apr-2024.)
(hadd(πœ‘, πœ“, πœ’) ↔ hadd(πœ“, πœ’, πœ‘))
 
Theoremwl-3xorcoma 36663 Commutative law for triple xor. Copy of hadcoma 1599. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
(hadd(πœ‘, πœ“, πœ’) ↔ hadd(πœ“, πœ‘, πœ’))
 
Theoremwl-3xorcomb 36664 Commutative law for triple xor. (Contributed by Mario Carneiro, 4-Sep-2016.) df-had redefined. (Revised by Wolf Lammen, 24-Apr-2024.)
(hadd(πœ‘, πœ“, πœ’) ↔ hadd(πœ‘, πœ’, πœ“))
 
Theoremwl-3xornot1 36665 Flipping the first input flips the triple xor. wl-3xorrot 36662 can rotate any input to the front, so flipping any one of them does the same. (Contributed by Wolf Lammen, 1-May-2024.)
(Β¬ hadd(πœ‘, πœ“, πœ’) ↔ hadd(Β¬ πœ‘, πœ“, πœ’))
 
Theoremwl-3xornot 36666 Triple xor distributes over negation. Copy of hadnot 1602. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)
(Β¬ hadd(πœ‘, πœ“, πœ’) ↔ hadd(Β¬ πœ‘, Β¬ πœ“, Β¬ πœ’))
 
Theoremwl-1xor 36667 In the recursive scheme

"(n+1)-xor" ↔ if-(πœ‘, Β¬ "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 βŠ₯, as a sequence of 0 inputs never has an odd number being part of it. (Contributed by Wolf Lammen, 11-May-2024.)

(if-(πœ“, Β¬ βŠ₯, βŠ₯) ↔ πœ“)
 
Theoremwl-2xor 36668 In the recursive scheme

"(n+1)-xor" ↔ if-(πœ‘, Β¬ "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 36667). (Contributed by Wolf Lammen, 11-May-2024.)

(if-(πœ‘, Β¬ πœ“, πœ“) ↔ (πœ‘ ⊻ πœ“))
 
Theoremwl-df-3mintru2 36669 Alternative definition of wcad 1606. See df-cad 1607 to learn how it is currently introduced. The only use case so far is being a binary addition primitive for df-sad 16397. 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 1607 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 1087, and "(at least) 3 out of 3", expressed by triple conjunction df-3an 1088.

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 βŠ₯, because a sequence of zero inputs never has a positive number of them true. The general case adds a new input πœ‘ 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-(πœ‘, "n-mintru-m" , "n-mintru-(m+1)" )

Our definition here matches "3-mintru-2" with inputs πœ‘, πœ“ and πœ’. Starting from the base cases we find after simplifications: "2-mintru-2" (πœ“, πœ’) ↔ (πœ“ ∧ πœ’) (wl-2mintru2 36676), and "2-mintru-1" (πœ“, πœ’) ↔ (πœ“ ∨ πœ’) (wl-2mintru1 36675). 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(πœ‘, πœ“, πœ’) ↔ if-(πœ‘, (πœ“ ∨ πœ’), (πœ“ ∧ πœ’)))
 
Theoremwl-df2-3mintru2 36670 The adder carry in disjunctive normal form. An alternative highly symmetric definition emphasizing the independence of order of the inputs πœ‘, πœ“ and πœ’. Copy of cador 1608. (Contributed by Mario Carneiro, 4-Sep-2016.) df-cad redefined. (Revised by Wolf Lammen, 12-Jun-2024.)
(cadd(πœ‘, πœ“, πœ’) ↔ ((πœ‘ ∧ πœ“) ∨ (πœ‘ ∧ πœ’) ∨ (πœ“ ∧ πœ’)))
 
Theoremwl-df3-3mintru2 36671 The adder carry in conjunctive normal form. An alternative highly symmetric definition emphasizing the independence of order of the inputs πœ‘, πœ“ and πœ’. Copy of cadan 1609. (Contributed by Mario Carneiro, 4-Sep-2016.) df-cad redefined. (Revised by Wolf Lammen, 18-Jun-2024.)
(cadd(πœ‘, πœ“, πœ’) ↔ ((πœ‘ ∨ πœ“) ∧ (πœ‘ ∨ πœ’) ∧ (πœ“ ∨ πœ’)))
 
Theoremwl-df4-3mintru2 36672 An alternative definition of the adder carry. Copy of df-cad 1607. (Contributed by Mario Carneiro, 4-Sep-2016.) df-cad redefined. (Revised by Wolf Lammen, 19-Jun-2024.)
(cadd(πœ‘, πœ“, πœ’) ↔ ((πœ‘ ∧ πœ“) ∨ (πœ’ ∧ (πœ‘ ⊻ πœ“))))
 
Theoremwl-1mintru1 36673 Using the recursion formula:

"(n+1)-mintru-(m+1)" ↔ if-(πœ‘, "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 ⊀ / βŠ₯.

Negating an "n-mintru1" operation means: All n inputs πœ‘.. πœƒ are false. This is also conveniently expressed as Β¬ (πœ‘ ∨.. ∨ πœƒ). 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-(πœ’, ⊀, βŠ₯) ↔ πœ’)
 
Theoremwl-1mintru2 36674 Using the recursion formula:

"(n+1)-mintru-(m+1)" ↔ if-(πœ‘, "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 βŠ₯.

Since no sequence of inputs has a longer subsequence of whatever property, the resulting βŠ₯ 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 (πœ‘ ⊼ πœ“), 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-(πœ’, βŠ₯, βŠ₯) ↔ βŠ₯)
 
Theoremwl-2mintru1 36675 Using the recursion formula

"(n+1)-mintru-(m+1)" ↔ if-(πœ‘, "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 36673 shows "1-mintru-1" is equivalent to the only input.

Negating an "n-mintru1" operation means: All n inputs πœ‘.. πœƒ are false. This is also conveniently expressed as Β¬ (πœ‘ ∨.. ∨ πœƒ), in accordance with the result here. (Contributed by Wolf Lammen, 10-May-2024.)

(if-(πœ“, ⊀, πœ’) ↔ (πœ“ ∨ πœ’))
 
Theoremwl-2mintru2 36676 Using the recursion formula

"(n+1)-mintru-(m+1)" ↔ if-(πœ‘, "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 36673 and wl-1mintru2 36674 to see that "1-mintru-1" / "1-mintru-2" evaluate to πœ’ / βŠ₯ 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 (πœ‘ ⊼ πœ“), 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-(πœ“, πœ’, βŠ₯) ↔ (πœ“ ∧ πœ’))
 
Theoremwl-df3maxtru1 36677 Assuming "(n+1)-maxtru1" ↔ Β¬ "(n+1)-mintru-2", we can deduce from the recursion formula given in wl-df-3mintru2 36669, that a similiar one

"(n+1)-maxtru1" ↔ if-(πœ‘,-. "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 πœ‚, 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.)

(Β¬ cadd(πœ‘, πœ“, πœ’) ↔ if-(πœ‘, (πœ“ ⊽ πœ’), (πœ“ ⊼ πœ’)))
 
21.20.5  An alternative axiom ~ ax-13
 
Axiomax-wl-13v 36678* A version of ax13v 2371 with a distinctor instead of a distinct variable condition.

Had we additionally required π‘₯ and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1912. 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.)

(Β¬ βˆ€π‘₯ π‘₯ = 𝑦 β†’ (𝑦 = 𝑧 β†’ βˆ€π‘₯ 𝑦 = 𝑧))
 
Theoremwl-ax13lem1 36679* A version of ax-wl-13v 36678 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.)
(Β¬ βˆ€π‘₯ π‘₯ = 𝑦 β†’ (𝑧 = 𝑦 β†’ βˆ€π‘₯ 𝑧 = 𝑦))
 
21.20.6  Other stuff
 
Theoremwl-mps 36680 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.)
(πœ‘ β†’ (πœ“ β†’ πœ’))    &   ((πœ‘ β†’ πœ’) β†’ πœƒ)    β‡’   ((πœ‘ β†’ πœ“) β†’ πœƒ)
 
Theoremwl-syls1 36681 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.)
(πœ“ β†’ πœ’)    &   ((πœ‘ β†’ πœ’) β†’ πœƒ)    β‡’   ((πœ‘ β†’ πœ“) β†’ πœƒ)
 
Theoremwl-syls2 36682 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.)
(πœ‘ β†’ πœ“)    &   ((πœ‘ β†’ πœ’) β†’ πœƒ)    β‡’   ((πœ“ β†’ πœ’) β†’ πœƒ)
 
Theoremwl-embant 36683 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.)
πœ‘    &   (πœ“ β†’ πœ’)    β‡’   ((πœ‘ β†’ πœ“) β†’ πœ’)
 
Theoremwl-orel12 36684 In a conjunctive normal form a pair of nodes like (πœ‘ ∨ πœ“) ∧ (Β¬ πœ‘ ∨ πœ’) eliminates the need of a node (πœ“ ∨ πœ’). This theorem allows simplifications in that respect. (Contributed by Wolf Lammen, 20-Jun-2020.)
(((πœ‘ ∨ πœ“) ∧ (Β¬ πœ‘ ∨ πœ’)) β†’ (πœ“ ∨ πœ’))
 
Theoremwl-cases2-dnf 36685 A particular instance of orddi 1007 and anddi 1008 converting between disjunctive and conjunctive normal forms, when both πœ‘ and Β¬ πœ‘ appear. This theorem in fact rephrases cases2 1045, and is related to consensus 1050. I restate it here in DNF and CNF. The proof deliberately does not use df-ifp 1061 and dfifp4 1064, by which it can be shortened. (Contributed by Wolf Lammen, 21-Jun-2020.) (Proof modification is discouraged.)
(((πœ‘ ∧ πœ“) ∨ (Β¬ πœ‘ ∧ πœ’)) ↔ ((Β¬ πœ‘ ∨ πœ“) ∧ (πœ‘ ∨ πœ’)))
 
Theoremwl-cbvmotv 36686* 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.)
(βˆƒ*π‘₯⊀ β†’ βˆƒ*π‘¦βŠ€)
 
Theoremwl-moteq 36687 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.)
(βˆƒ*π‘₯⊀ β†’ 𝑦 = 𝑧)
 
Theoremwl-motae 36688 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.)
(βˆƒ*π‘’βŠ€ β†’ βˆ€π‘₯ 𝑦 = 𝑧)
 
Theoremwl-moae 36689* 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 1970 in the proof, that ensures "at least one thing exists". For other equivalences see wl-euae 36690 and exists1 2655. Gerard Lang pointed out, that βˆƒπ‘¦βˆ€π‘₯π‘₯ = 𝑦 with disjoint π‘₯ and 𝑦 (df-mo 2533, trut 1546) 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 βˆƒ*. (Revised by Wolf Lammen, 7-Mar-2023.)
(βˆƒ*π‘₯⊀ ↔ βˆ€π‘₯ π‘₯ = 𝑦)
 
Theoremwl-euae 36690* Two ways to express "exactly one thing exists" . (Contributed by Wolf Lammen, 5-Mar-2023.)
(βˆƒ!π‘₯⊀ ↔ βˆ€π‘₯ π‘₯ = 𝑦)
 
Theoremwl-nax6im 36691* The following series of theorems are centered around the empty domain, where no set exists. As a consequence, a set variable like π‘₯ has no instance to assign to. An expression like π‘₯ = 𝑦 is not really meaningful then. What does it evaluate to, true or false? In fact, the grammar extension weq 1965 requires us to formally assign a boolean value to an equation, say always false, unless you want to give up on exmid 892, for example. Whatever it is, we start out with the contraposition of ax-6 1970, 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.)
(Β¬ βˆƒπ‘₯ π‘₯ = 𝑦 β†’ πœ‘)    β‡’   (Β¬ βˆƒπ‘₯⊀ β†’ πœ‘)
 
Theoremwl-hbae1 36692 This specialization of hbae 2429 does not depend on ax-11 2153. (Contributed by Wolf Lammen, 8-Aug-2021.)
(βˆ€π‘₯ π‘₯ = 𝑦 β†’ βˆ€π‘¦βˆ€π‘₯ π‘₯ = 𝑦)
 
Theoremwl-naevhba1v 36693* An instance of hbn1w 2048 applied to equality. (Contributed by Wolf Lammen, 7-Apr-2021.)
(Β¬ βˆ€π‘₯ π‘₯ = 𝑦 β†’ βˆ€π‘₯ Β¬ βˆ€π‘₯ π‘₯ = 𝑦)
 
Theoremwl-spae 36694 Prove an instance of sp 2175 from ax-13 2370 and Tarski's FOL only, without distinct variable conditions. The antecedent βˆ€π‘₯π‘₯ = 𝑦 holds in a multi-object universe only if 𝑦 is substituted for π‘₯, or vice versa, i.e. both variables are effectively the same. The converse Β¬ βˆ€π‘₯π‘₯ = 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 βˆ€π‘₯π‘₯ = 𝑦 and Β¬ βˆ€π‘₯π‘₯ = 𝑦 can help eliminating distinct variable conditions.

The antecedent βˆ€π‘₯π‘₯ = 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 2170.

Note that this theorem is also provable from ax-12 2170 alone, so you can pick the axiom it is based on.

Compare this result to 19.3v 1984 and spaev 2054 having distinct variable conditions, but a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 5-Apr-2021.)

(βˆ€π‘₯ π‘₯ = 𝑦 β†’ π‘₯ = 𝑦)
 
Theoremwl-speqv 36695* Under the assumption Β¬ π‘₯ = 𝑦 a specialized version of sp 2175 is provable from Tarski's FOL and ax13v 2371 only. Note that this reverts the implication in ax13lem1 2372, so in fact (Β¬ π‘₯ = 𝑦 β†’ (βˆ€π‘₯𝑧 = 𝑦 ↔ 𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
(Β¬ π‘₯ = 𝑦 β†’ (βˆ€π‘₯ 𝑧 = 𝑦 β†’ 𝑧 = 𝑦))
 
Theoremwl-19.8eqv 36696* Under the assumption Β¬ π‘₯ = 𝑦 a specialized version of 19.8a 2173 is provable from Tarski's FOL and ax13v 2371 only. Note that this reverts the implication in ax13lem2 2374, so in fact (Β¬ π‘₯ = 𝑦 β†’ (βˆƒπ‘₯𝑧 = 𝑦 ↔ 𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
(Β¬ π‘₯ = 𝑦 β†’ (𝑧 = 𝑦 β†’ βˆƒπ‘₯ 𝑧 = 𝑦))
 
Theoremwl-19.2reqv 36697* Under the assumption Β¬ π‘₯ = 𝑦 the reverse direction of 19.2 1979 is provable from Tarski's FOL and ax13v 2371 only. Note that in conjunction with 19.2 1979 in fact (Β¬ π‘₯ = 𝑦 β†’ (βˆ€π‘₯𝑧 = 𝑦 ↔ βˆƒπ‘₯𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.)
(Β¬ π‘₯ = 𝑦 β†’ (βˆƒπ‘₯ 𝑧 = 𝑦 β†’ βˆ€π‘₯ 𝑧 = 𝑦))
 
Theoremwl-nfalv 36698* If π‘₯ is not present in πœ‘, it is not free in βˆ€π‘¦πœ‘. (Contributed by Wolf Lammen, 11-Jan-2020.)
β„²π‘₯βˆ€π‘¦πœ‘
 
Theoremwl-nfimf1 36699 An antecedent is irrelevant to a not-free property, if it always holds. I used this variant of nfim 1898 in dvelimdf 2447 to simplify the proof. (Contributed by Wolf Lammen, 14-Oct-2018.)
(βˆ€π‘₯πœ‘ β†’ (β„²π‘₯(πœ‘ β†’ πœ“) ↔ β„²π‘₯πœ“))
 
Theoremwl-nfae1 36700 Unlike nfae 2431, this specialized theorem avoids ax-11 2153. (Contributed by Wolf Lammen, 26-Jun-2019.)
β„²π‘₯βˆ€π‘¦ 𝑦 = π‘₯
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