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Theorem List for Metamath Proof Explorer - 31701-31800   *Has distinct variable group(s)
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Statement

21.14  Mathbox for BJ

In this mathbox, we try to respect the ordering of the sections of the main part. There are strengthenings of theorems of the main part, as well as work on reducing axiom dependencies.

21.14.1  Propositional calculus

Miscellaneous utility theorems of propositional calculus.

21.14.1.1  Derived rules of inference

In this section, we prove a few rules of inference derived from modus ponens, and which do not depend on any axioms.

Theorembj-mp2c 31701 A double modus ponens inference. (Contributed by BJ, 24-Sep-2019.)
𝜑    &   (𝜑𝜓)    &   (𝜑 → (𝜓𝜒))       𝜒

Theorembj-mp2d 31702 A double modus ponens inference. (Contributed by BJ, 24-Sep-2019.)
𝜑    &   (𝜑𝜓)    &   (𝜓 → (𝜑𝜒))       𝜒

21.14.1.2  A syntactic theorem

In this section, we prove a syntactic theorem (bj-0 31703) asserting that some formula is well-formed. Then, we use this syntactic theorem to shorten the proof of a "usual" theorem (bj-1 31704) and explain in the comment of that theorem why this phenomenon is unusual.

Theorembj-0 31703 A syntactic theorem. See the section comment and the comment of bj-1 31704. The full proof (that is, with the syntactic, non-essential steps) does not appear on this webpage. It has five steps and reads \$= wph wps wi wch wi \$. The only other syntactic theorems in the main part of set.mm are wel 1978 and weq 1861. (Contributed by BJ, 24-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
wff ((𝜑𝜓) → 𝜒)

Theorembj-1 31704 In this proof, the use of the syntactic theorem bj-0 31703 allows to reduce the total length by one (non-essential) step. See also the section comment and the comment of bj-0 31703. Since bj-0 31703 is used in a non-essential step, this use does not appear on this webpage (but the present theorem appears on the webpage for bj-0 31703 as a theorem referencing it). The full proof reads \$= wph wps wch bj-0 id \$. (while, without using bj-0 31703, it would read \$= wph wps wi wch wi id \$.).

Now we explain why syntactic theorems are not useful in set.mm. Suppose that the syntactic theorem thm-0 proves that PHI is a well-formed formula, and that thm-0 is used to shorten the proof of thm-1. Assume that PHI does have proper non-atomic subformulas (which is not the case of the formula proved by weq 1861 or wel 1978). Then, the proof of thm-1 does not construct all the proper non-atomic subformulas of PHI (if it did, then using thm-0 would not shorten it). Therefore, thm-1 is a special instance of a more general theorem with essentially the same proof. In the present case, bj-1 31704 is a special instance of id 22. (Contributed by BJ, 24-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

(((𝜑𝜓) → 𝜒) → ((𝜑𝜓) → 𝜒))

21.14.1.3  Minimal implicational calculus

Theorembj-a1k 31705 Weakening of ax-1 6. This shortens the proofs of dfwe2 6873, ordunisuc2 6936, r111 8521, smo11 7348. (Contributed by BJ, 11-Aug-2020.)
(𝜑 → (𝜓 → (𝜒𝜓)))

Theorembj-jarri 31706 Inference associated with jarr 104. Its associated inference is bj-jarrii 31707. (Contributed by BJ, 29-Mar-2020.)
((𝜑𝜓) → 𝜒)       (𝜓𝜒)

Theorembj-jarrii 31707 Inference associated with bj-jarri 31706. (Contributed by BJ, 29-Mar-2020.)
((𝜑𝜓) → 𝜒)    &   𝜓       𝜒

Theorembj-imim2ALT 31708 More direct proof of imim2 56. Note that imim2i 16 and imim2d 55 can be proved as usual from this closed form (i.e., using ax-mp 5 and syl 17 respectively). (Contributed by BJ, 19-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))

Theorembj-imim21 31709 The propositional function (𝜒 → (. → 𝜃)) is decreasing. (Contributed by BJ, 19-Jul-2019.)
((𝜑𝜓) → ((𝜒 → (𝜓𝜃)) → (𝜒 → (𝜑𝜃))))

Theorembj-imim21i 31710 Inference associated with bj-imim21 31709. Its associated inference is syl5 33. (Contributed by BJ, 19-Jul-2019.)
(𝜑𝜓)       ((𝜒 → (𝜓𝜃)) → (𝜒 → (𝜑𝜃)))

21.14.1.4  Positive calculus

Theorembj-orim2 31711 Proof of orim2 882 from the axiomatic definition of disjunction (olc 398, orc 399, jao 533) and minimal implicational calculus. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))

Theorembj-curry 31712 A non-intuitionistic positive statement, sometimes called a paradox of material implication. Sometimes called Curry's axiom. (Contributed by BJ, 4-Apr-2021.)
(𝜑 ∨ (𝜑𝜓))

Theorembj-peirce 31713 Proof of peirce 192 from minimal implicational calculus, the axiomatic definition of disjunction (olc 398, orc 399, jao 533), and Curry's axiom bj-curry 31712. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜑) → 𝜑)

Theorembj-currypeirce 31714 Curry's axiom (a non-intuitionistic statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 192 over minimal implicational calculus and the axiomatic definition of disjunction (olc 398, orc 399, jao 533). A shorter proof from bj-orim2 31711, pm1.2 534, syl6com 36 is possible if we accept to use pm1.2 534, itself a direct consequence of jao 533. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∨ (𝜑𝜓)) → (((𝜑𝜓) → 𝜑) → 𝜑))

Theorembj-peircecurry 31715 Peirce's axiom peirce 192 implies Curry's axiom over minimal implicational calculus and the axiomatic definition of disjunction (olc 398, orc 399, jao 533). See comment of bj-currypeirce 31714. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ∨ (𝜑𝜓))

21.14.1.5  Implication and negation

Theorempm4.81ALT 31716 Alternate proof of pm4.81 380. (Contributed by BJ, 30-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑𝜑) ↔ 𝜑)

Theorembj-con4iALT 31717 Alternate proof of con4i 112. Probably the original proof. (Contributed by BJ, 29-Mar-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑 → ¬ 𝜓)       (𝜓𝜑)

Theorembj-con2com 31718 A commuted form of the contrapositive, true in minimal calculus. (Contributed by BJ, 19-Mar-2020.)
(𝜑 → ((𝜓 → ¬ 𝜑) → ¬ 𝜓))

Theorembj-con2comi 31719 Inference associated with bj-con2com 31718. Its associated inference is mt2 190. TODO: when in the main part, add to mt2 190 that it is the inference associated with bj-con2comi 31719. (Contributed by BJ, 19-Mar-2020.)
𝜑       ((𝜓 → ¬ 𝜑) → ¬ 𝜓)

Theorembj-pm2.01i 31720 Inference associated with pm2.01 179. (Contributed by BJ, 30-Mar-2020.)
(𝜑 → ¬ 𝜑)        ¬ 𝜑

Theorembj-nimn 31721 If a formula is true, then it does not imply its negation. (Contributed by BJ, 19-Mar-2020.) A shorter proof is possible using id 22 and jc 158, however, the present proof uses theorems that are more basic than jc 158. (Proof modification is discouraged.)
(𝜑 → ¬ (𝜑 → ¬ 𝜑))

Theorembj-nimni 31722 Inference associated with bj-nimn 31721. (Contributed by BJ, 19-Mar-2020.)
𝜑        ¬ (𝜑 → ¬ 𝜑)

Theorembj-peircei 31723 Inference associated with peirce 192. (Contributed by BJ, 30-Mar-2020.)
((𝜑𝜓) → 𝜑)       𝜑

Theorembj-looinvi 31724 Inference associated with looinv 193. Its associated inference is bj-looinvii 31725. (Contributed by BJ, 30-Mar-2020.)
((𝜑𝜓) → 𝜓)       ((𝜓𝜑) → 𝜑)

Theorembj-looinvii 31725 Inference associated with bj-looinvi 31724. (Contributed by BJ, 30-Mar-2020.)
((𝜑𝜓) → 𝜓)    &   (𝜓𝜑)       𝜑

21.14.1.6  Disjunction

A few lemmas about disjunction. The fundamental theorems in this family are the dual statements pm4.71 660 and pm4.72 916. See also biort 936 and biorf 419.

Theorembj-jaoi1 31726 Shortens 11 proofs by a total of around 60 bytes. (Contributed by BJ, 30-Sep-2019.)
(𝜑𝜓)       ((𝜑𝜓) → 𝜓)

Theorembj-jaoi2 31727 Shortens 9 proofs by a total of around 50 bytes. (Contributed by BJ, 30-Sep-2019.)
(𝜑𝜓)       ((𝜓𝜑) → 𝜓)

21.14.1.7  Logical equivalence

A few other characterizations of the bicondional. The inter-definability of logical connectives offers many ways to express a given statement. Some useful theorems in this regard are df-or 384, df-an 385, pm4.64 386, imor 427, pm4.62 434 through pm4.67 443, and, for the De Morgan laws, ianor 508 through pm4.57 517.

Theorembj-dfbi4 31728 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Theorembj-dfbi5 31729 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) ↔ ((𝜑𝜓) → (𝜑𝜓)))

Theorembj-dfbi6 31730 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))

Theorembj-bijust0 31731 The general statement that bijust 194 proves (with a shorter proof). (Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Revised by BJ, 19-Mar-2020.)
¬ ((𝜑𝜑) → ¬ (𝜑𝜑))

21.14.1.8  The conditional operator for propositions

Theorembj-consensus 31732 Version of consensus 990 expressed using the conditional operator. (Remark: it may be better to express it as consensus 990, using only binary connectives, and hinting at the fact that it is a Boolean algebra identity, like the absorption identities.) (Contributed by BJ, 30-Sep-2019.)
((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))

Theorembj-consensusALT 31733 Alternate proof of bj-consensus 31732. (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))

Theorembj-dfifc2 31734* This should be the alternate definition of "ifc" if "if-" enters the main part. (Contributed by BJ, 20-Sep-2019.)
if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵))}

Theorembj-df-ifc 31735* The definition of "ifc" if "if-" enters the main part. This is in line with the definition of a class as the extension of a predicate in df-clab 2597. (Contributed by BJ, 20-Sep-2019.)
if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)}

Theorembj-ififc 31736* A theorem linking if- and if. (Contributed by BJ, 24-Sep-2019.)
(𝑥 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑥𝐴, 𝑥𝐵))

21.14.1.9  Propositional calculus: miscellaneous

Miscellaneous theorems of propositional calculus.

Theoremsylancl2 31737 Shortens 5 proofs. (Contributed by BJ, 25-Apr-2019.)
(𝜑𝜓)    &   𝜒    &   ((𝜓𝜒) ↔ 𝜃)       (𝜑𝜃)

Theoremsylancl3 31738 Shortens 11 proofs by a total of around 150 bytes. (Contributed by BJ, 25-Apr-2019.)
(𝜑𝜓)    &   𝜒    &   (𝜃 ↔ (𝜓𝜒))       (𝜑𝜃)

Theorembj-imbi12 31739 Imported form (uncurried form) of imbi12 335. (Contributed by BJ, 6-May-2019.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) ↔ (𝜓𝜃)))

Theorembj-trut 31740 A proposition is equivalent to it being implied by . Closed form of trud 1484 (which it can shorten); dual of dfnot 1493. It is to tbtru 1485 what a1bi 351 is to tbt 358, and this appears in their respective proofs. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
(𝜑 ↔ (⊤ → 𝜑))

Theorembj-biorfi 31741 This should be labeled "biorfi" while the current biorfi 421 should be labeled "biorfri". The dual of biorf 419 is not biantr 968 but iba 523 (and ibar 524). So there should also be a "biorfr". (Note that these four statements can actually be strengthened to biconditionals.) (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
¬ 𝜑       (𝜓 ↔ (𝜑𝜓))

Theorembj-falor 31742 Dual of truan 1492 (which has biconditional reversed). (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
(𝜑 ↔ (⊥ ∨ 𝜑))

Theorembj-falor2 31743 Dual of truan 1492. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
((⊥ ∨ 𝜑) ↔ 𝜑)

Theorembj-bibibi 31744 A property of the biconditional. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
(𝜑 ↔ (𝜓 ↔ (𝜑𝜓)))

Theorembj-imn3ani 31745 Duplication of bnj1224 30126. Three-fold version of imnani 438. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by BJ, 22-Oct-2019.) (Proof modification is discouraged.)
¬ (𝜑𝜓𝜒)       ((𝜑𝜓) → ¬ 𝜒)

Theorembj-andnotim 31746 Two ways of expressing a certain ternary connective. Note the respective positions of the three formulas on each side of the biconditional. (Contributed by BJ, 6-Oct-2018.)
(((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))

Theorembj-bi3ant 31747 This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.)
(𝜑 → (𝜓𝜒))       (((𝜃𝜏) → 𝜑) → (((𝜏𝜃) → 𝜓) → ((𝜃𝜏) → 𝜒)))

Theorembj-bisym 31748 This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.)
(((𝜑𝜓) → (𝜒𝜃)) → (((𝜓𝜑) → (𝜃𝜒)) → ((𝜑𝜓) → (𝜒𝜃))))

21.14.2  Modal logic

In this section, we prove some theorems related to modal logic. For modal logic, we refer to https://en.wikipedia.org/wiki/Kripke_semantics, https://en.wikipedia.org/wiki/Modal_logic and https://plato.stanford.edu/entries/logic-modal/.

Monadic first-order logic (i.e., with quantification over only one variable) is bi-interpretable with modal logic, by mapping 𝑥 to "necessity" (generally denoted by a box) and 𝑥 to "possibility" (generally denoted by a diamond). Therefore, we use these quantifiers so as not to introduce new symbols. (To be strictly within modal logic, we should add dv conditions between 𝑥 and any other metavariables appearing in the statements.)

For instance, ax-gen 1713 corresponds to the necessitation rule of modal logic, and ax-4 1728 corresponds to the distributivity axiom (K) of modal logic, also called the Kripke scheme. Modal logics satisfying these rule and axiom are called "normal modal logics", of which the most important modal logics are.

The minimal normal modal logic is also denoted by (K). Here are a few normal modal logics with their axiomatizations (on top of (K)): (K) axiomatized by no supplementary axioms; (T) axiomatized by the axiom T; (K4) axiomatized by the axiom 4; (S4) axiomatized by the axioms T,4; (S5) axiomatized by the axioms T,5 or D,B,4; (GL) axiomatized by the axiom GL.

The last one, called Gödel–Löb logic or provability logic, is important because it describes exactly the properties of provability in Peano arithmetic, as proved by Robert Solovay. See for instance https://plato.stanford.edu/entries/logic-provability/. A basic result in this logic is bj-gl4 31753.

Theorembj-axdd2 31749 This implication, proved using only ax-gen 1713 and ax-4 1728 on top of propositional calculus (hence holding, up to the standard interpretation, in any normal modal logic), shows that the axiom scheme 𝑥 implies the axiom scheme (∀𝑥𝜑 → ∃𝑥𝜑). These correspond to the modal axiom (D), and in predicate calculus, they assert that the universe of discourse is nonempty. For the converse, see bj-axd2d 31750. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))

Theorembj-axd2d 31750 This implication, proved using only ax-gen 1713 on top of propositional calculus (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme (∀𝑥𝜑 → ∃𝑥𝜑) implies the axiom scheme 𝑥. These correspond to the modal axiom (D), and in predicate calculus, they assert that the universe of discourse is nonempty. For the converse, see bj-axdd2 31749. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥⊤ → ∃𝑥⊤) → ∃𝑥⊤)

Theorembj-axtd 31751 This implication, proved from propositional calculus only (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme (∀𝑥𝜑𝜑) (modal T) implies the axiom scheme (∀𝑥𝜑 → ∃𝑥𝜑) (modal D). See also bj-axdd2 31749 and bj-axd2d 31750. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → ((∀𝑥𝜑𝜑) → (∀𝑥𝜑 → ∃𝑥𝜑)))

Theorembj-gl4lem 31752 Lemma for bj-gl4 31753. Note that this proof holds in the modal logic (K). (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑 → ∀𝑥(∀𝑥(∀𝑥𝜑𝜑) → (∀𝑥𝜑𝜑)))

Theorembj-gl4 31753 In a normal modal logic, the modal axiom GL implies the modal axiom (4). Note that the antecedent of bj-gl4 31753 is an instance of the axiom GL, with 𝜑 replaced by (∀𝑥𝜑𝜑), sometimes called the "strong necessity" of 𝜑. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥(∀𝑥(∀𝑥𝜑𝜑) → (∀𝑥𝜑𝜑)) → ∀𝑥(∀𝑥𝜑𝜑)) → (∀𝑥𝜑 → ∀𝑥𝑥𝜑))

Theorembj-axc4 31754 Over minimal calculus, the modal axiom (4) (hba1 2137) and the modal axiom (K) (ax-4 1728) together imply axc4 2115. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥𝜑 → ∀𝑥𝑥𝜑) → ((∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝑥𝜑 → ∀𝑥𝜓)) → (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))))

21.14.3  Provability logic

In this section, we assume that, on top of propositional calculus, there is given a provability predicate Prv satisfying the three axioms ax-prv1 31756 and ax-prv2 31757 and ax-prv3 31758. Note the similarity with ax-gen 1713, ax-4 1728 and hba1 2137 respectively. These three properties of Prv are often called the Hilbert–Bernays–Löb derivability conditions, or the Hilbert–Bernays provability conditions.

This corresponds to the modal logic (K4) (see previous section for modal logic). The interpretation of provability logic is the following: we are given a background first-order theory T, the wff Prv 𝜑 means "𝜑 is provable in T", and the turnstile indicates provability in T.

Beware that "provability logic" often means (K) augmented with the Gödel–Löb axiom GL, which we do not assume here (at least for the moment). See for instance https://plato.stanford.edu/entries/logic-provability/.

Provability logic is worth studying because whenever T is a first-order theory containing Robinson arithmetic (a fragment of Peano arithmetic), one can prove (using Gödel numbering, and in the much weaker primitive recursive arithmetic) that there exists in T a provability predicate Prv satisfying the above three axioms. (We do not construct this predicate in this section; this is still a project.)

The main theorems of this section are the "easy parts" of the proofs of Gödel's second incompleteness theorem (bj-babygodel 31761) and Löb's theorem (bj-babylob 31762). See the comments of these theorems for details.

Syntaxcprvb 31755 Syntax for the provability predicate.
wff Prv 𝜑

Axiomax-prv1 31756 First property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
𝜑       Prv 𝜑

Axiomax-prv2 31757 Second property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(Prv (𝜑𝜓) → (Prv 𝜑 → Prv 𝜓))

Axiomax-prv3 31758 Third property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(Prv 𝜑 → Prv Prv 𝜑)

Theoremprvlem1 31759 An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(𝜑𝜓)       (Prv 𝜑 → Prv 𝜓)

Theoremprvlem2 31760 An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(𝜑 → (𝜓𝜒))       (Prv 𝜑 → (Prv 𝜓 → Prv 𝜒))

The first hypothesis reads "𝜑 is true if and only if it is not provable in T" (and having this first hypothesis means that we can prove this fact in T). The wff 𝜑 is a formal version of the sentence "This sentence is not provable". The hard part of the proof of Gödel's theorem is to construct such a 𝜑, called a "Gödel–Rosser sentence", for a first-order theory T which is effectively axiomatizable and contains Robinson arithmetic, through Gödel diagonalization (this can be done in primitive recursive arithmetic). The second hypothesis means that is not provable in T, that is, that the theory T is consistent (and having this second hypothesis means that we can prove in T that the theory T is consistent). The conclusion is the falsity, so having the conclusion means that T can prove the falsity, that is, T is inconsistent.

Therefore, taking the contrapositive, this theorem expresses that if a first-order theory is consistent (and one can prove in it that some formula is true if and only if it is not provable in it), then this theory does not prove its own consistency.

This proof is due to George Boolos, Gödel's Second Incompleteness Theorem Explained in Words of One Syllable, Mind, New Series, Vol. 103, No. 409 (January 1994), pp. 1--3.

(Contributed by BJ, 3-Apr-2019.)

(𝜑 ↔ ¬ Prv 𝜑)    &    ¬ Prv ⊥

Theorembj-babylob 31762 See the section header comments for the context, as well as the comments for bj-babygodel 31761.

Löb's theorem when the Löb sentence is given as a hypothesis (the hard part of the proof of Löb's theorem is to construct this Löb sentence; this can be done, using Gödel diagonalization, for any first-order effectively axiomatizable theory containing Robinson arithmetic). More precisely, the present theorem states that if a first-order theory proves that the provability of a given sentence entails its truth (and if one can construct in this theory a provability predicate and a Löb sentence, given here as the first hypothesis), then the theory actually proves that sentence.

See for instance, Eliezer Yudkowsky, The Cartoon Guide to Löb's Theorem (available at http://yudkowsky.net/rational/lobs-theorem/).

(Contributed by BJ, 20-Apr-2019.)

(𝜓 ↔ (Prv 𝜓𝜑))    &   (Prv 𝜑𝜑)       𝜑

Theorembj-godellob 31763 Proof of Gödel's theorem from Löb's theorem (see comments at bj-babygodel 31761 and bj-babylob 31762 for details). (Contributed by BJ, 20-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ↔ ¬ Prv 𝜑)    &    ¬ Prv ⊥

21.14.4  First-order logic

Utility lemmas or strengthenings of theorems in the main part (biconditional or closed forms, or fewer dv conditions, or dv conditions replaced with non-freeness hypotheses...). Sorted in the same order as in the main part.

21.14.4.1  Universal and existential quantifiers, "non-free" predicate

Syntaxwnff 31764 Syntax for the ℲℲ predicate.
wff ℲℲ𝑥𝜑

Definitiondf-bj-nf 31765 Alternate definition of the "semantically non-free" predicate, equivalent to nf5 2102 by df-nf 1701. This definition is stricter than nf5 2102 as soon as one has sp 2041, and less strict as soon as one has ax-10 2006. This version has no nested quantifiers, so is easier to understand and easier to handle when ax-10 2006 is not yet available, as illustrated by the results below (see bj-nfbi 31793 bj-nfn 31795). (Contributed by BJ, 6-May-2019.)
(ℲℲ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))

Theorembj-nf2 31766 Alternate definition of df-bj-nf 31765. (Contributed by BJ, 6-May-2019.)
(ℲℲ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))

Theorembj-nf3 31767 Alternate definition of df-bj-nf 31765. (Contributed by BJ, 6-May-2019.)
(ℲℲ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))

Theorembj-nf4 31768 Alternate definition of df-bj-nf 31765. This definition uses only primitive symbols. (Contributed by BJ, 6-May-2019.)
(ℲℲ𝑥𝜑 ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))

Theorembj-nftht 31769 Closed form of nfth 1718. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝜑 → ℲℲ𝑥𝜑)

Theorembj-nfntht 31770 Closed form of nfnth 1719. (Contributed by BJ, 6-May-2019.)
(¬ ∃𝑥𝜑 → ℲℲ𝑥𝜑)

Theorembj-nfntht2 31771 Closed form of nfnth 1719. (Contributed by BJ, 6-May-2019.)
(∀𝑥 ¬ 𝜑 → ℲℲ𝑥𝜑)

Theorembj-nfth 31772 Any variable is not free in a theorem. (Contributed by BJ, 6-May-2019.)
𝜑       ℲℲ𝑥𝜑

Theorembj-nftru 31773 The true constant has no free variables. (Contributed by BJ, 6-May-2019.)
ℲℲ𝑥

Theorembj-nfnth 31774 Any variable is not free in a falsity. (Contributed by BJ, 6-May-2019.)
¬ 𝜑       ℲℲ𝑥𝜑

Theorembj-nffal 31775 The false constant has no free variables. (Contributed by BJ, 6-May-2019.)
ℲℲ𝑥

Theorembj-genr 31776 Generalization rule on the right conjunct. See 19.28 2083. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (𝜑 ∧ ∀𝑥𝜓)

Theorembj-genl 31777 Generalization rule on the left conjunct. See 19.27 2082. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theorembj-genan 31778 Generalization rule on a conjunction. Forward inference associated with 19.26 1786. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (∀𝑥𝜑 ∧ ∀𝑥𝜓)

Theorembj-2alim 31779 Closed form of 2alimi 1731. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∀𝑥𝑦𝜑 → ∀𝑥𝑦𝜓))

Theorembj-2exim 31780 Closed form of 2eximi 1753. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 → ∃𝑥𝑦𝜓))

Theorembj-alanim 31781 Closed form of alanimi 1734. (Contributed by BJ, 6-May-2019.)
(∀𝑥((𝜑𝜓) → 𝜒) → ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒))

Theorembj-2albi 31782 Closed form of 2albii 1738. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦𝜓))

Theorembj-notalbii 31783 Equivalence of universal quantification of negation of equivalent formulas. Shortens ab0 3905, ballotlem2 29877, bnj1143 30115, hausdiag 21258. (Contributed by BJ, 17-Jul-2021.)
(𝜑𝜓)       (∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ 𝜓)

Theorembj-2exbi 31784 Closed form of 2exbii 1765. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝜓))

Theorembj-3exbi 31785 Closed form of 3exbii 1766. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦𝑧(𝜑𝜓) → (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓))

Theorembj-sylgt2 31786 Uncurried form of sylgt 1739. (Contributed by BJ, 2-May-2019.)
((∀𝑥(𝜓𝜒) ∧ (𝜑 → ∀𝑥𝜓)) → (𝜑 → ∀𝑥𝜒))

Theorembj-exlimh 31787 Closed form of close to exlimih 2133. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑𝜓) → ((∃𝑥𝜓𝜒) → (∃𝑥𝜑𝜒)))

Theorembj-exlimh2 31788 Uncurried form of bj-exlimh 31787. (Contributed by BJ, 2-May-2019.)
((∀𝑥(𝜑𝜓) ∧ (∃𝑥𝜓𝜒)) → (∃𝑥𝜑𝜒))

Theorembj-alrimhi 31789 An inference associated with sylgt 1739 and bj-exlimh 31787. (Contributed by BJ, 12-May-2019.)
(𝜑𝜓)       (ℲℲ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜓))

Theorembj-nexdh 31790 Closed form of nexdh 1779 (and more general since it uses 𝜒). (Contributed by BJ, 6-May-2019.)
(∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓)))

Theorembj-nexdh2 31791 Uncurried form of bj-nexdh 31790. (Contributed by BJ, 6-May-2019.)
((∀𝑥(𝜑 → ¬ 𝜓) ∧ (𝜒 → ∀𝑥𝜑)) → (𝜒 → ¬ ∃𝑥𝜓))

Theorembj-hbxfrbi 31792 Closed form of hbxfrbi 1742. Notes: it is less important than bj-nfbi 31793; it requires sp 2041 (unlike bj-nfbi 31793); there is an obvious version with (∃𝑥𝜑𝜑) instead. (Contributed by BJ, 6-May-2019.)
(∀𝑥(𝜑𝜓) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))

Theorembj-nfbi 31793 Closed form of nfbii 1770 (with df-bj-nf 31765 instead of nf5 2102, which would require more axioms). (Contributed by BJ, 6-May-2019.)
(∀𝑥(𝜑𝜓) → (ℲℲ𝑥𝜑 ↔ ℲℲ𝑥𝜓))

Theorembj-nfxfr 31794 Proof of nfxfr 1771 from bj-nfbi 31793. (Contributed by BJ, 6-May-2019.)
(𝜑𝜓)    &   ℲℲ𝑥𝜑       ℲℲ𝑥𝜓

Theorembj-nfn 31795 A variable is non-free in a proposition if and only if it is so in its negation. Requires fewer axioms than nfn 1768. (Contributed by BJ, 6-May-2019.)
(ℲℲ𝑥𝜑 ↔ ℲℲ𝑥 ¬ 𝜑)

Theorembj-exlime 31796 Variant of exlimih 2133 where the non-freeness of 𝑥 in 𝜓 is expressed using an existential quantifier. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝜓𝜓)    &   (𝜑𝜓)       (∃𝑥𝜑𝜓)

Theorembj-exnalimn 31797 A transformation of quantifiers and logical connectives. The general statement that equs3 1862 proves.

This and the following theorems are the general instances of already proved theorems. They could be moved to the main part, before ax-5 1827. I propose to move to the main part: bj-exnalimn 31797, bj-exaleximi 31800, bj-exalimi 31801, bj-ax12i 31803, bj-ax12wlem 31807, bj-ax12w 31852, and remove equs3 1862. A new label is needed for bj-ax12i 31803 and label suggestions are welcome for the others. I also propose to change ¬ ∀𝑥¬ to 𝑥 in speimfw 1863 and spimfw 1865 (other spim* theorems use 𝑥 and very very few theorems in set.mm use ¬ ∀𝑥¬). (Contributed by BJ, 29-Sep-2019.)

(∃𝑥(𝜑𝜓) ↔ ¬ ∀𝑥(𝜑 → ¬ 𝜓))

Theorembj-nalnaleximiOLD 31798 An inference for distributing quantifiers over a double implication. The general statement that speimfw 1863 proves. (Contributed by BJ, 12-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜒 → (𝜑𝜓))       (¬ ∀𝑥 ¬ 𝜒 → (∀𝑥𝜑 → ∃𝑥𝜓))

Theorembj-nalnalimiOLD 31799 An inference for distributing quantifiers over a double implication. The general statement that spimfw 1865 proves. (Contributed by BJ, 12-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜒 → (𝜑𝜓))    &   𝜓 → ∀𝑥 ¬ 𝜓)       (¬ ∀𝑥 ¬ 𝜒 → (∀𝑥𝜑𝜓))

Theorembj-exaleximi 31800 An inference for distributing quantifiers over a double implication. (Almost) the general statement that speimfw 1863 proves. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))       (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))

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