P. 375 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Axiom

ax-luk1 37401

1 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-1 1651 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 37402

2 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-2 1652 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 pm2.65 193):

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

⊢ ((¬ 𝜑 → 𝜑) → 𝜑)

Axiom

ax-luk3 37403

3 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-3 1653 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 37401, ax-luk2 37402 and pm2.21 123 result in phi1 or phi2, meaning they always hold. But for wl-luk-ax3 37415 we can find a counter example resulting in phi0, not a statement always true. The verification of a particular set of axioms in a given model is tedious and error prone, so I wrote a computer program, first checking this for me, and second, hunting for a counter example. Here is the result, after 9165 fruitlessly computer generated models:

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

Negation is defined as
----------------------------------------------------------------------

-. phi0	-. phi1	-. phi2
phi1	phi0	phi1

Implication is defined as
----------------------------------------------------------------------

p->q	q: phi0	q: phi1	q: phi2
p: phi0	phi1	phi1	phi1
p: phi1	phi0	phi1	phi1
p: phi2	phi0	phi0	phi0

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

⊢ (𝜑 → (¬ 𝜑 → 𝜓))

21.22.1  1. Bootstrapping

Theorem

wl-section-boot 37404

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 37405

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 37406

An inference version of the transitive laws for implication luk-1 1651. 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 37407

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 37408

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 37409

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 37410

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.)

⊢ 𝜑    ⇒   ⊢ (¬ 𝜑 → 𝜓)

Theorem

wl-luk-a1i 37411

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 37412

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 37413

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 37414

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 37415

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 37416

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 37417

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 37418

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 37419

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 37420

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.)

⊢ (¬ 𝜑 → 𝜓)    ⇒   ⊢ (¬ 𝜓 → 𝜑)

Theorem

wl-luk-ja 37421

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.)

⊢ (¬ 𝜑 → 𝜒)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ ((𝜑 → 𝜓) → 𝜒)

Theorem

wl-luk-imim2 37422

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 37423

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 37424

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 37425

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 37426

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 130 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 37427

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)))

21.22.2  Implication chains

Theorem

wl-section-impchain 37428

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 37429

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 37430

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 37431

This theorem is in fact a copy of wl-luk-syl 37406, 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 37432

This theorem is in fact a copy of wl-luk-imtrdi 37414, 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 37433

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 37429 series developed before. (Contributed by Wolf Lammen, 17-Nov-2019.)

⊢ ⊤

Theorem

wl-impchain-com-1.1 37434

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 37435, 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 37435

This theorem is in fact a copy of wl-luk-com12 37418, 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 37433 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 37436

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 37433 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 37437

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 37433 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 37438

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 37433 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 37433 series. (Contributed by Wolf Lammen, 17-Nov-2019.)

⊢ ⊤

Theorem

wl-impchain-com-2.3 37439

This theorem is in fact a copy of com23 86. It starts a series of theorems named after wl-impchain-com-n.m 37438. 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 37440

This theorem is in fact a copy of com24 95. It is another instantiation of theorems named after wl-impchain-com-n.m 37438. 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 37441

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 37442

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 37411, that allows prepending an arbitrary antecedent to an implication chain. Using our antecedent swapping theorems described in wl-impchain-com-n.m 37438, 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 depends 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 37433 series developed before. (Contributed by Wolf Lammen, 20-Jun-2020.)

⊢ ⊤

Theorem

wl-impchain-a1-1 37443

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 37444

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 37445

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)))

21.22.3  Theorems around the conditional operator

Theorem

wl-ifp-ncond1 37446

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 37447

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 37448

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 37449

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)))

21.22.4  Alternative development of hadd, cadd

Theorem

wl-df-3xor 37450

Alternative definition of whad 1589 based on hadifp 1601. See df-had 1590 to learn how it is currently introduced. The only use case so far is being a binary addition primitive for df-sad 16484. 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 37465).

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 1590, 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 37451

Alternative definition of wl-df-3xor 37450, using triple exclusive disjunction, or XOR3. You can add more input by appending each one with a ⊻. Copy of hadass 1593. (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 37452

Alternative form of wl-df3xor2 37451. Copy of df-had 1590. (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 37453

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 37454

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 37455

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 37456

Alternative form of wl-3xorbi 37455. (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 37457

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 37458

Equivalence theorem for triple xor. Copy of hadbi123i 1592. (Contributed by Mario Carneiro, 4-Sep-2016.)

⊢ (𝜓 ↔ 𝜒)    &   ⊢ (𝜃 ↔ 𝜏)    &   ⊢ (𝜂 ↔ 𝜁)    ⇒   ⊢ (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁))

Theorem

wl-3xorrot 37459

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 37460

Commutative law for triple xor. Copy of hadcoma 1595. (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 37461

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 37462

Flipping the first input flips the triple xor. wl-3xorrot 37459 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 37463

Triple xor distributes over negation. Copy of hadnot 1598. (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 37464

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 37465

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 37464). (Contributed by Wolf Lammen, 11-May-2024.)

⊢ (if-(𝜑, ¬ 𝜓, 𝜓) ↔ (𝜑 ⊻ 𝜓))

Theorem

wl-df-3mintru2 37466

Alternative definition of wcad 1602. See df-cad 1603 to learn how it is currently introduced. The only use case so far is being a binary addition primitive for df-sad 16484. 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 1603 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 ⊥, 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 37473), and "2-mintru-1" (𝜓, 𝜒) ↔ (𝜓 ∨ 𝜒) (wl-2mintru1 37472). 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 37467

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 1604. (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 37468

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 1605. (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 37469

An alternative definition of the adder carry. Copy of df-cad 1603. (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 37470

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 equivalent 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 37471

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 37472

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 37470 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 37473

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 37470 and wl-1mintru2 37471 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 37474

Assuming "(n+1)-maxtru1" ↔ ¬ "(n+1)-mintru-2", we can deduce from the recursion formula given in wl-df-3mintru2 37466, 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 simultaneously 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)))

21.22.5  An alternative axiom ~ ax-13

Axiom

ax-wl-13v 37475*

A version of ax13v 2375 with a distinctor instead of a distinct variable condition.

Had we additionally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1907. 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 37476*

A version of ax-wl-13v 37475 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. This proof bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 23-Jul-2021.)

⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

21.22.6  Bootstrapping set theory with classes

Theorem

wl-cleq-0 37477*

Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong. This and the following texts should be read as explorations rather than as definite statements, open to doubt, alternatives, and reinterpretation.

Preface

Three specific theorems are under focus in the following pages: df-cleq 2726, df-clel 2813, and df-clab 2712. Only technical concepts necessary to explain these will be introduced, along with a selection of supporting theorems. The three theorems are central to a bootstrapping process that introduces objects into set.mm. We will first examine how Metamath in general creates basic new ideas from scratch, and then look at how these methods are applied specifically to classes, capable of representing objects in set theory.

In Zermelo-Fraenkel set theory with the axiom of choice (ZFC), these three theorems are (more or less) independent of each other, which means they can be introduced in different orders. From my own experience, another order has pedagogical advantages: it helps grasping not only the overall concept better, but also the intricate details that I first found difficult to comprehend. Reordering theorems, though syntactically possible, sometimes may cause doubts when intermediate results are not strictly tied to ZFC only.

The purpose of set.mm is to provide a formal framework capable of proving the results of ZFC, provided that formulas are properly interpreted. In fact, there is freedom of interpretation. The database set.mm develops from the very beginning, where nothing is assumed or fixed, and gradually builts up to the full abstraction of ZFC. Along the way, results are only preliminary, and one may at any point branch off and pursue a different path toward another variant of set theory. This openess is already visible in axiom ax-mp 5, where the symbol → can be understood as as implication, bi-conditional, or conjunction. The notation and symbol shapes are suggestive, but their interpretation is not mandatory. The point here is that Metamath, as a purely syntactic system, can sometimes allow freedoms, unavailable to semantically fixed systems, which presuppose only a single ultimate goal.

(Contributed by Wolf Lammen, 28-Sep-2025.)

⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Theorem

wl-cleq-1 37478*

Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong. This and the following texts should be read as explorations rather than as definite statements, open to doubt, alternatives, and reinterpretation.

Grammars and Parsable contents

A Metamath program is a text-based tool. Its input consists of primarily of human-readable and editable text stored in *.mm files. For automated processing, these files follow a strict structure. This enables automated analysis of their contents, verification of proofs, proof assistance, and the generation of output files - for example, the HTML page of this dummy theorem. A set.mm file contains numerous such structured instructions serving these purposes.

This page provides a brief explanation of the general concepts behind structured text, as exemplified by set.mm. The study of structured text originated in linguistics, and later computer science formalized it further for text like data and program code. The rules describing such structures are collectively known as grammar. Metamath also introduces a grammar to support automation and establish a high degree of confidence in its correctness. It could have been described using the terminology of earlier scientific disciplines, but instead it uses its own language.

When a text exhibits a sufficiently regular structure, its form can be described by a set of syntax rules, or grammar. Such rules consist of terminal symbols (fixed, literal elements) and non-terminal symbols, which can recursively be expanded using the grammar's rewrite rules. A program component that applies a grammar to text is called a parser. The parser decomposes the text into smaller parts, called syntactic units, while often maintaining contextual information. These units may then be handed over to subsequent components for further processing, such as populating a database or generating output.

In these pages, we restrict our attention to strictly formatted material consisting of formulas with logical and mathematical content. These syntactic units are embedded in higher-level structures such as chapters, together with commands that, for example, control the HTML output.

Conceptually, the parsing process can be viewed as consisting of two stages. The top-level stage applies a simple built-in grammar to identify its structural units. Each unit is a text region marked on both sides with predefined tokens (in Metamath: keywords), beginning with the escape character "$". Text regions containing logical or mathematical formulas are then passed to a second-stage parser, which applies a different grammar. Unlike the first, this grammar is not built-in but is dynamically constructed.

In what follows, we will ignore the first stage of parsing, since its role is only to extract the relevant material embedded within text primarily intended for human readers.

(Contributed by Wolf Lammen, 18-Aug-2025.)

⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Theorem

wl-cleq-2 37479*

Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong. This and the following texts should be read as explorations rather than as definite statements, open to doubt, alternatives, and reinterpretation.

Introducing a New Concept: Well-formed formulas

The parser that processes strictly formatted Metamath text with logical or mathematical content constructs its grammar dynamically. New syntax rules are added on the fly whenever they are needed to parse upcoming formulas. Only a minimal set of built-in rules - those required to introduce new grammar rules - is predefined; everything else must be supplied by set.mm itself as syntactic units identified by the first-stage parser. Text regions beginning with the tokens "$c" or "$v", for example, are part of such grammar extensions.

We will outline the extension process used when introducing an entirely new concept that cannot build on any prior material. As a simple example, we will trace here the steps involved in defining formulas, the expressions used for hypotheses and statements - a concept first-order logic needs from the very beginning to articulate its ideas.

1. Introduce a type code

To mark new formulas and variables later used in theorems, the constant "wff" is reserved. It abbreviates "well-formed formula", a term that already suggests that such formulas must be syntactically valid and therefore parseable to enable automatic processing.

2. Introduce variables

Variables with unique names such as 𝜑, 𝜓, ..., intended to later represent formulas of type "wff". In grammar terms, they correspond to non-terminal symbols. These variables are marked with the type "wff". Variables are fundamental in Metamath, since they enable substitution during proofs: variables of a particular type can be consistently replaced by any formula of the same type. In grammar terms, this corresponds to applying a rewrite rule. At this step, however, no rewrite rule exists; we can only substitute one "wff" variable for another. This suffices only for very elementary theorems such as idi 1.

3. Add primitive formulas

Rewrite rules describing primitive formulas of type "wff" are then added to the grammar. Typically, they describe an operator (a constant in the grammar) applied to one or more variables, possibly of different types (e.g. ∀𝑥𝜑, although at this stage only "wff" is available). Since variables are non-terminal symbols, more complex formulas can be constructed from primitive ones, by consistently replacing variables with any wff formula - whether involving the same operator or different ones introduced by other rewrite rules. Whenever such a replacement introduces variables again, they may in turn recursively be replaced.

If an operator takes two variables of type wff, it is called a binary connective in logic. The first such operator encountered is (𝜑 → 𝜓). Based on its token, its intended meaning is material implication, though this interpretation is not fixed from the outset.

4. Specify the properties of primitive formulas

Once the biconditional connective is available for formulas, new connectives can be defined by specifying replacement formulas that rely solely on previously introduced material. Such definitions makes it possible to eliminate the definiens.

At the very beginning, however, this is not possible for well-formed formulas, since little or no prior material exists. Instead, the semantics of an expression such as (𝜑 → 𝜓) are progressively constrained by axioms - that is, theorems without proof. The first such axiom for material implication is ax-mp 5, with additional axioms following later.

(Contributed by Wolf Lammen, 21-Aug-2025.)

⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Theorem

wl-cleq-3 37480*

Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong. This and the following texts should be read as explorations rather than as definite statements, open to doubt, alternatives, and reinterpretation.

Introducing a New Concept: Classes

In wl-cleq-2 37479 we examined how the basic notion of a well-formed formula is introduced in set.mm. A similar process is used to add the notion of a class to Metamath. This process is somewhat more involved, since two parallel variants are established: sets and the broader notion of classes, which include sets (see 3a. below).

In Zermelo-Fraenkel set theory (ZF) classes will serve as a convenient shorthand that simplifies formulas and proofs. Ultimately, only sets - a part of all classes - are intended to exist as actual objects.

In the First Order Logic (FOL) portion of set.mm objects themselves are not used - only variables representing them. It is not even assumed that objects must be sets at all. In principle, the universe of discourse could consist of anything - vegetables, text strings, and so on. For this reason, the type code "setvar", used for object variables, is somewhat of a misnomer. Its final meaning - and the name that goes with it - becomes justified only in later developments.

We will now revisit the four basic steps presented in wl-cleq-2 37479, this time focusing on object variables and paying special attention to the additional complexities that arise from extending sets to classes.

1. Introduce type codes

1a. Reserve a type code for classes, specifically the grammar constant "class". Initially, this type code applies to class variables and will later, beyond these four steps, also be assigned to formulas that define specific classes, i.e. objects.

1b. Reserve a type code for set variables, represented by the grammar constant "setvar". The name itself indicates that this type code will never be assigned to a formula describing a specific set, but only to variables containing such objects.

2. Introduce variables

2a. Use variables with unique names such as 𝐴, 𝐵, ..., to represent classes. These class variables are assigned the type code "class", ensuring that only formulas of type "class" can be substituted for them during proofs.

2b. Use variables with unique names such as 𝑎, 𝑏 , ..., to represent sets. These variables are assigned the type code "setvar". During a substitution in a proof, a variable of type "setvar" may only be replaced with another variable of this type. No specific formula, or object, of this type exists.

3. Add primitive formulas

3a. Add the rewrite rule cv 1535 to set.mm, allowing variables of type "setvar" to also aquire type "class". In this way, a variable of "setvar" type can serve as a substitute for a class variable.

3b. Add the rewrite rules wcel 2105 and wceq 1536 to set.mm, making the formulas 𝐴 ∈ 𝐵 and 𝐴 = 𝐵 valid well-formed formulas (wff). Since variables of type "setvar" can be substituted for class variables, 𝑥 ∈ 𝑦 and 𝑥 = 𝑦 are also provably valid wff.

In the FOL part of set.mm, only these specific formulas play a role and are therefore treated as primitive there. The underlying universe may not contain sets, and the notion of a class is not even be required.

Additional mixed-type formulas, such as 𝑥 ∈ 𝐴, 𝐴 ∈ 𝑥, 𝑥 = 𝐴, and 𝐴 = 𝑥 exist. When the theory is later refined to distinguish between sets and classes, the results from FOL remain valid and naturally extend to these mixed cases. These cases will occasionally be examined individually in subsequent discussions.

4. Specify the properties of primitive formulas

In FOL in set.mm, the formulas 𝑥 = 𝑦 and 𝑥 ∈ 𝑦 cannot be derived from earlier material, and therefore cannot be defined. Instead, their fundamental properties are established through axioms, namely ax-6 1964 through ax-9 2115, ax-13 2374, and ax-ext 2705.

Similarly, axioms establish the properties of the primitive formulas 𝐴 = 𝐵 and 𝐴 ∈ 𝐵, ensuring that they extend the FOL counterparts 𝑥 = 𝑦 and 𝑥 ∈ 𝑦 in a consistent and meaningful way.

At the same time, a criterion must be developed to distinguish sets from classes. Since set variables can only be substituted by other set variables, equality must permit the assignment of class terms known to represent sets to those variables.

(Contributed by Wolf Lammen, 25-Aug-2025.)

⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Theorem

wl-cleq-4 37481*

Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong. This and the following texts should be read as explorations rather than as definite statements, open to doubt, alternatives, and reinterpretation.

Semantics of Equality

In https://us.metamath.org/mpeuni/mmset.html#class some basic information is given about class variables, and what they are intended to represent. They range over so called class builders, i.e. definitions with type code "class". At this stage, only one such builder is explicitly available: cv 1535. Zermelo-Fraenkel set theory (ZF) introduces just one more in df-clab 2712.

Restricting ourselves to exactly these class builders is not necessary for fixing the general properties of equality between classes. Other models of set theory can still be supported at this point - even with an unlimited number of class builders. In line with the step-by-step approach of gradually closing in on ZF set theory, we introduce axioms of limited complexity, each addressing only a small number of properties.

1. **General properties of equality**

There is a broadly shared understanding of what equality between objects expresses, extending beyond mathematics or set theory. Equality must at least form an equivalence relation, where 𝑥, 𝑦, and 𝑧 are objects of the universe under consideration::

1a. Reflexivity 𝑥 = 𝑥

1b. Symmetry (𝑥 = 𝑦 → 𝑦 = 𝑥

1c. Transitivity ((𝑥 = 𝑦 ∧ 𝑥 = 𝑧) → 𝑦 = 𝑧)

In addition equality must obey the Identity of Indiscernables (Leibniz's Law). It states that distinct, or unequal, objects cannot share all the same properties. In set.mm properties are expressed by formulas. Thus equal objects must always behave the same, yielding the same results when substituted in formulas.

2. **Equality in First Order Logic (FOL)**

Let us now see how FOL captures these principles. Several axioms center on equality, namely ax-6 1964, ax-7 2004, ax-8 2107, ax-9 2115 and ax-12 2174, and ax-13 . In paractice, restricted versions with distinct variable conditions are used: ax6v 1965, ax12v 2175. The unrestricted axioms together with axiom ax-13 2374 allow elimination of distinct variable conditions, but this gain is considered too minor to justify their routine use.

Equality in FOL is formalized as follows:

2a. Equality as an equivalence relation is essentially covered by ax-7 2004.

2b. For the primitive ∈ operator, Leibniz's Law is captured by ax-8 2107 and ax-9 2115.

2c. Its general form is stated in sbequ12 2248.

2d. Implicit substitution: assuming Leibniz's Law for a particular expression, there are theorems that extend its validity to other, derived expressions. Frequently, the derived expression introduces quantification over the original one.

3. **Equality between classes**

We require Leibniz's Law to hold for equality between classes. For a mixed expression 𝑥 = 𝐴 the sethood of 𝐴 should follow from that of 𝑥. Thus, the equation 𝑥 = 𝐴 can be regarded as a criterion for the sethood of 𝐴. In ZF, where only sets exist, this is expressed instead by ∃𝑥𝑥 = 𝐴 stating the existence of 𝐴 in the same fashion that the restricted axiom ax6ev 1966 does for set variables.

Equality between classes must still be an equivalence relation. It must also be conservative, meaning that the results described in 2. remain valid when both class variables in 𝐴 = 𝐵 are replaced by set variables. Assuming class variables are all sets, we require that ax6ev 1966, ax-7 , ax-8 , ax-9 and ax12v2 2176 are provable when free set variables are replaced by class variables. In this way, the assignment 𝑥 = 𝐴 can stand in for the syntactically impossible substitution of a set variable by a class variable, even if the class is known to contain a set.

Since in ZF we cannot quantify over a class variables, no pure class-based version of the quantified FOL axioms exist. Moreover, in ZF classes are not considered an integral part of the theory, so they need to be eliminable. This also impacts the treatment of equality if it is regarded as a primitive operation.

(Contributed by Wolf Lammen, 18-Sep-2025.)

⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Axiom

ax-wl-cleq 37482*

Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong.

This text should be read as an exploration rather than as a definite statement, open to doubt, alternatives, and reinterpretation.

At the point where df-cleq 2726 is introduced, the foundations of set theory are being established through the notion of a class. A central property of classes is what elements, expressed by the membership operator ∈, belong to them . Quantification (∀𝑥) applies only to objects that a variable of kind setvar can represent. These objects will henceforth be called sets. Some classes may not be sets; these are called proper classes. It remains open at this stage whether membership can involve them.

The formula given in df-cleq 2726 (restated below) asserts that two classes are equal if and only if they have exactly the same sets as elements. If proper classes are also admitted as elements, then two equal classes could still differ by such elements, potentially violating Leibniz's Law. A future axiom df-clel 2813 addresses this issue; df-cleq 2726 alone does not.

**Primitive connectives and class builders**

Specially crafted primitive operators on classes or class builders could introduce properties of classes beyond membership, not reflected in the formula here. This again risks violating Leibniz's Law. Therefore, the introduction of any future primitive operator or class builder must include a conservativity check to ensure consistency with Leibniz's Law.

**This axiom covers only some principles of equality**

The notion of equality expressed in this axiom does not automatically coincide with the general notion of equality. Some principles are, however, already captured: Equality is shown to be an equivalence relation, covering transivity (eqtr 2757), reflexivity (eqid 2734) and symmetry (eqcom 2741). It also yields the class-level version of ax-ext 2705 (the backward direction of df-cleq 2726) holds.

If we assume 𝑥 = 𝐴 holds, then substituting the free set variable 𝑦 with 𝐴 in ax6ev 1966 and ax12v2 2176 yields provable theorems (see wl-isseteq 37485, and wl-ax12v2cl 37486). However, a bound variable cannot be replaced with a class variable, since quantification over classes is not permitted. Taken together with the results from the previous paragraph, this shows that a class variable equal to a set behaves the same as a set variable, provided it is not quantified.

**Conservativity**

Moreover, this axiom is already partly derivable if all class variables are replaced by variables of type "setvar". In that case, the statement reduces to an instance of axextb 2708. This shows that the class builder cv 1535 is consistent with this axiom.

**Eliminable operator**

Finally, this axiom supports the idea that proper classes, and operators between them, should be eliminable, as required by ZF: It reduces equality to their membership properties. However, since the term 𝑥 ∈ 𝐴 is still undefined, elimination reduces equality to just something not yet clarified.

**Axiom vs Definition**

Up to this point, the only content involving class variables comes from the syntax definitions wceq 1536 and wcel 2105. Axioms are therefore required to progressively refine the semantics of classes until provable results coincide with our intended conception of set theory. This refinement process is explained in Step 4 of wl-cleq-2 37479.

From this perspective, df-cleq 2726 is in fact an axiom in disguise and would more appropriately be named ax-cleq.

At first glance, one might think that 𝐴 = 𝐵 is defined by the right-hand side of the biconditional. This would make 𝑥 ∈ 𝐴, i.e. membership of a set in a class, the more primitive concept, from which equality of classes could be derived. Such a viewpoint would be coherent if the properties of membership could be fully determined by other axioms. In my (WL's') opinion, however, the more direct and fruitful approach is not to construct class equality from membership, but to treat equality itself as axiomatic.

(Contributed by Wolf Lammen, 25-Aug-2025.)

⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Axiom

ax-wl-clel 37483*

Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong.

This text should be read as an exploration rather than as a definite statement, open to doubt, alternatives, and reinterpretation.

The formula in df-clel 2813 (restated below) states that only those classes for which ∃𝑥𝑥 = 𝐴 holds can be members of classes. Thus, a member of a class is always equal to a set, which excludes proper classes from class membership.

As explained in wl-cleq-4 37481, item 3, ∃𝑥𝑥 = 𝐴 is a sufficient criterion for a class to be a set, provided that Leibniz's Law holds for equality. Therefore this axiom is often rephrased as: classes contain only sets as members.

**Principles of equality**

Using this axiom we can derive the class-level counterparts of ax-8 2107 (see eleq2 2827) and ax-9 2115 (see eleq1 2826). Since ax-wl-cleq 37482 already asserts that equality between classes is an equivalence relation, the operators = and ∈ alone cannot distinguish equal classes. Hence, if membership is the only property that matters for classes, Leibniz's Law will hold. Later, however, additional class builders may introduce further properties of classes. A conservativity check for such builders can ensure this does not occur.

**Eliminability**

If we replace the class variable 𝐴 with a set variable 𝑧 in this axiom, the auxiliary variable 𝑥 can be eliminated, leaving only the trivial result (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵). Thus, df-clel 2813 by itself does not determine when a set is a member of a class. From this perspective, df-clel 2813 is in fact an axiom in disguise and would more appropriately be called ax-clel.

Overall, our axiomization leaves the meaning of fundamental expressions 𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵 open. All other fundamental formulas of set theory (𝐴 not a set variable, 𝐴 ∈ 𝐵, 𝑥 = 𝐵 𝐴 = 𝐵) can be reduced solely to the basic formulas 𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵.

If an axiomatization leaves a fundamental formula like 𝑥 ∈ 𝐴 unspecified, we could in principle define it bi-conditionally by any formula whatsoever - for example, the trivial ⊤. This, however, is not the approach we take. Instead, an appropriate class builder such as df-clab 2712 fills this gap.

(Contributed by Wolf Lammen, 26-Aug-2025.)

⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

Theorem

wl-df-clab 37484

Disclaimer: The material presented here is just my (WL's) personal perception. I am not an expert in this field, so some or all of the text here can be misleading, or outright wrong.

This text should be seen as an exploration, rather than viewing it as set in stone, no doubt or alternatives possible.

We now introduce the notion of class abstraction, which allows us to describe a specific class, in contrast to class variables that can stand for any class indiscriminately.

A new syntactic form is introduced for class abstractions, {𝑦 ∣ 𝜑}, read as "the class of sets 𝑦 such that 𝜑(𝑦)". This form is assigned the type "class" in cab 2711, so it can consistently substitute for a class variable during the syntactic construction process.

**Eliminability**

The axioms ax-wl-cleq 37482 and ax-wl-clel 37483 leave only 𝑥 ∈ 𝐴 unspecified. The definition of this class builder directly corresponds to that expression. When a class abstraction replaces the variable 𝐴 and 𝐵, then 𝐴 = 𝐵 and 𝐴 ∈ 𝐵 can be expressed in terms of these abstractions.

For general eliminability two conditions are needed:

1. Any class builder must replace 𝑥 ∈ 𝐴 with an expression containing no class variables. If necessary, class variables must be eliminated via a finite recursive process.

2. There must only be finitely many class builders. If a class variable could range over infinitely many builders, eliminability would fail, since unknown future builders would always need to be considered.

Condition (2) is met in set.mm by defining no class builder beyond cv 1535 and df-clab 2712. Thus we may assume that a class variable represents either a set variable, or a class abstraction:

a. If it represents a set variable, substitution eliminates it immediately.

b. If it equals a set variable 𝑥, then by cvjust 2728 it can be replaced with {𝑦 ∣ 𝑦 ∈ 𝑥}.

c. If it represents a proper class, then it equals some abstraction {𝑥 ∣ 𝜑}. If 𝜑 contains no class variables, elimination using 𝜑 is possible. The same holds if finite sequence of elimination steps renders 𝜑 free of class variables.

d. It represents a proper class, but 𝜑 in {𝑥 ∣ 𝜑} still contains non-eliminable class variables, then eliminability fails. A simple example is {𝑥 ∣ 𝑥 ∈ 𝐴}. Class variables can only appear in fundamental expressions 𝐴 = 𝐵 or 𝐴 ∈ 𝐵, Both can be reduced to forms involving 𝑧 ∈ 𝐴. Thus, in the expression 𝑧 ∈ {𝑥 ∣ 𝑥 ∈ 𝐴}, we still must eliminate 𝐴. Applying df-clab 2712 reduces it back to 𝑧 ∈ 𝐴, returning us to the starting point.

Case (d) shows that in full generality, a class variable cannot always be eliminated, something Zermelo-Fraenkel set theory (ZF) requires. If the universe contained only finitely many sets, a free class variable 𝐴 could be expressed as a finite disjunction of possiblities, hence eliminable. But in ZF's richer universe, in a definition of an unrestricted class variable 𝐴 = {𝑥 ∣ 𝜑} the variable 𝜑 will contain 𝐴 in some way, violating condition (1) above. Thus constraints are needed. In ZF, any formula containing class variables assumes that non-set class variables can be be replaced by {𝑥 ∣ 𝜑} where 𝜑 itself contains no class variables. There is, however, no way to state this condition in a formal way in set.mm.

Class abstractions themselves, however, can be eliminated, so df-clab is a definition.

**Definition checker**

How can case (d) be avoided? A solution is to restrict generality: require that in the definition of any concrete class abstraction {𝑥 ∣ 𝜑}, the formula 𝜑 is either free of class variables or built only from previously defined constructions. Such a restriction could be part of the definition checker.

In practice, the Metamath definition checker requires definitions to follow the specific pattern "⊢ {𝑥 ∣ 𝜑} = ...". Although df-clab 2712 does not conform to this pattern, it nevertheless permits elimination of class abstractions. Eliminability is the essential property of a valid definition, so df-clab 2712 can legitimately be regarded as one.

For further material on the elimination of class abstractions, see BJ's work beginning with eliminable1 36841 and one comment in https://github.com/metamath/set.mm/pull/4971.

(Contributed by Wolf Lammen, 28-Aug-2025.)

⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

Theorem

wl-isseteq 37485*

A class equal to a set variable implies it is a set. Note that 𝐴 may be dependent on 𝑥. The consequent, resembling ax6ev 1966, is the accepted expression for the idea of a class being a set. Sometimes a simpler expression like the antecedent here, or in elisset 2820, is already sufficient to mark a class variable as a set. (Contributed by Wolf Lammen, 7-Sep-2025.)

⊢ (𝑥 = 𝐴 → ∃𝑦 𝑦 = 𝐴)

Theorem

wl-ax12v2cl 37486*

The class version of ax12v2 2176, where the set variable 𝑦 is replaced with the class variable 𝐴. This is possible if 𝐴 is known to be a set, expressed by the antecedent.

Theorem ax12v 2175 is a specialization of ax12v2 2176. So any proof using ax12v 2175 will still hold if ax12v2 2176 is used instead.

Theorem ax12v2 2176 expresses that two equal set variables cannot be distinguished by whatever complicated formula 𝜑 if one is replaced with the other in it. This theorem states a similar result for a class variable known to be a set: All sets equal to the class variable behave the same if they replace the class variable in 𝜑.

Most axioms in FOL containing an equation correspond to a theorem where a class variable known to be a set replaces a set variable in the formula. Some exceptions cannot be avoided: The set variable must nowhere be bound. And it is not possible to state a distinct variable condition where a class 𝐴 is different from another, or distinct from a variable with type wff. So ax-12 2174 proper is out of reach: you cannot replace 𝑦 in ∀𝑦𝜑 with a class variable.

But where such limitations are not violated, the proof of the FOL theorem should carry over to a version where a class variable, known to be set, appears instead of a set variable. (Contributed by Wolf Lammen, 8-Aug-2020.)

⊢ (∃𝑦 𝑦 = 𝐴 → (𝑥 = 𝐴 → (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))

21.22.7  Other stuff

Theorem

wl-mps 37487

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 37488

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 37489

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 37490

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 37491

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 37492

A particular instance of orddi 1011 and anddi 1012 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 37493*

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 37494

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 37495

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 37496*

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 1964 in the proof, that ensures "at least one thing exists". For other equivalences see wl-euae 37497 and exists1 2658. Gerard Lang pointed out, that ∃𝑦∀𝑥𝑥 = 𝑦 with disjoint 𝑥 and 𝑦 (df-mo 2537, trut 1542) 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 37497*

Two ways to express "exactly one thing exists" . (Contributed by Wolf Lammen, 5-Mar-2023.)

⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

Theorem

wl-nax6im 37498*

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 1959 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 1964, 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 37499

This specialization of hbae 2433 does not depend on ax-11 2154. (Contributed by Wolf Lammen, 8-Aug-2021.)

⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦)

Theorem

wl-naevhba1v 37500*

An instance of hbn1w 2043 applied to equality. (Contributed by Wolf Lammen, 7-Apr-2021.)

⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)