P. 318 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
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.
Theorem	bj-mp2c 31701	A double modus ponens inference. (Contributed by BJ, 24-Sep-2019.)
⊢ 𝜑    &   ⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ 𝜒
Theorem	bj-mp2d 31702	A double modus ponens inference. (Contributed by BJ, 24-Sep-2019.)
⊢ 𝜑    &   ⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 → (𝜑 → 𝜒))    ⇒   ⊢ 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.
Theorem	bj-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 ((𝜑 → 𝜓) → 𝜒)
Theorem	bj-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.)
⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜑 → 𝜓) → 𝜒))
21.14.1.3  Minimal implicational calculus
Theorem	bj-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 → (𝜒 → 𝜓)))
Theorem	bj-jarri 31706	Inference associated with jarr 104. Its associated inference is bj-jarrii 31707. (Contributed by BJ, 29-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜒)    ⇒   ⊢ (𝜓 → 𝜒)
Theorem	bj-jarrii 31707	Inference associated with bj-jarri 31706. (Contributed by BJ, 29-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜒)    &   ⊢ 𝜓    ⇒   ⊢ 𝜒
Theorem	bj-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.)
⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))
Theorem	bj-imim21 31709	The propositional function (𝜒 → (. → 𝜃)) is decreasing. (Contributed by BJ, 19-Jul-2019.)
⊢ ((𝜑 → 𝜓) → ((𝜒 → (𝜓 → 𝜃)) → (𝜒 → (𝜑 → 𝜃))))
Theorem	bj-imim21i 31710	Inference associated with bj-imim21 31709. Its associated inference is syl5 33. (Contributed by BJ, 19-Jul-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜒 → (𝜓 → 𝜃)) → (𝜒 → (𝜑 → 𝜃)))
21.14.1.4  Positive calculus
Theorem	bj-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.)
⊢ ((𝜑 → 𝜓) → ((𝜒 ∨ 𝜑) → (𝜒 ∨ 𝜓)))
Theorem	bj-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 ∨ (𝜑 → 𝜓))
Theorem	bj-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.)
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
Theorem	bj-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 ∨ (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
Theorem	bj-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 ∨ (𝜑 → 𝜓))
21.14.1.5  Implication and negation
Theorem	pm4.81ALT 31716	Alternate proof of pm4.81 380. (Contributed by BJ, 30-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((¬ 𝜑 → 𝜑) ↔ 𝜑)
Theorem	bj-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)    ⇒   ⊢ (𝜓 → 𝜑)
Theorem	bj-con2com 31718	A commuted form of the contrapositive, true in minimal calculus. (Contributed by BJ, 19-Mar-2020.)
⊢ (𝜑 → ((𝜓 → ¬ 𝜑) → ¬ 𝜓))
Theorem	bj-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)
Theorem	bj-pm2.01i 31720	Inference associated with pm2.01 179. (Contributed by BJ, 30-Mar-2020.)
⊢ (𝜑 → ¬ 𝜑)    ⇒   ⊢ ¬ 𝜑
Theorem	bj-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))
Theorem	bj-nimni 31722	Inference associated with bj-nimn 31721. (Contributed by BJ, 19-Mar-2020.)
⊢ 𝜑    ⇒   ⊢ ¬ (𝜑 → ¬ 𝜑)
Theorem	bj-peircei 31723	Inference associated with peirce 192. (Contributed by BJ, 30-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜑)    ⇒   ⊢ 𝜑
Theorem	bj-looinvi 31724	Inference associated with looinv 193. Its associated inference is bj-looinvii 31725. (Contributed by BJ, 30-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜓)    ⇒   ⊢ ((𝜓 → 𝜑) → 𝜑)
Theorem	bj-looinvii 31725	Inference associated with bj-looinvi 31724. (Contributed by BJ, 30-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜓)    &   ⊢ (𝜓 → 𝜑)    ⇒   ⊢ 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.
Theorem	bj-jaoi1 31726	Shortens 11 proofs by a total of around 60 bytes. (Contributed by BJ, 30-Sep-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜑 ∨ 𝜓) → 𝜓)
Theorem	bj-jaoi2 31727	Shortens 9 proofs by a total of around 50 bytes. (Contributed by BJ, 30-Sep-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜓 ∨ 𝜑) → 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.
Theorem	bj-dfbi4 31728	Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
Theorem	bj-dfbi5 31729	Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)))
Theorem	bj-dfbi6 31730	Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∧ 𝜓)))
Theorem	bj-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.)
⊢ ¬ ((𝜑 → 𝜑) → ¬ (𝜑 → 𝜑))
21.14.1.8  The conditional operator for propositions
Theorem	bj-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) ∨ (𝜓 ∧ 𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))
Theorem	bj-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) ∨ (𝜓 ∧ 𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))
Theorem	bj-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 ∣ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))}
Theorem	bj-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, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}
Theorem	bj-ififc 31736*	A theorem linking if- and if. (Contributed by BJ, 24-Sep-2019.)
⊢ (𝑥 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵))
21.14.1.9  Propositional calculus: miscellaneous Miscellaneous theorems of propositional calculus.
Theorem	sylancl2 31737	Shortens 5 proofs. (Contributed by BJ, 25-Apr-2019.)
⊢ (𝜑 → 𝜓)    &   ⊢ 𝜒    &   ⊢ ((𝜓 ∧ 𝜒) ↔ 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	sylancl3 31738	Shortens 11 proofs by a total of around 150 bytes. (Contributed by BJ, 25-Apr-2019.)
⊢ (𝜑 → 𝜓)    &   ⊢ 𝜒    &   ⊢ (𝜃 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	bj-imbi12 31739	Imported form (uncurried form) of imbi12 335. (Contributed by BJ, 6-May-2019.)
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))
Theorem	bj-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))
Theorem	bj-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 ↔ (𝜑 ∨ 𝜓))
Theorem	bj-falor 31742	Dual of truan 1492 (which has biconditional reversed). (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
⊢ (𝜑 ↔ (⊥ ∨ 𝜑))
Theorem	bj-falor2 31743	Dual of truan 1492. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
⊢ ((⊥ ∨ 𝜑) ↔ 𝜑)
Theorem	bj-bibibi 31744	A property of the biconditional. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
⊢ (𝜑 ↔ (𝜓 ↔ (𝜑 ↔ 𝜓)))
Theorem	bj-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.)
⊢ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜒)
Theorem	bj-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) ↔ ((𝜑 → 𝜓) ∨ 𝜒))
Theorem	bj-bi3ant 31747	This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (((𝜃 → 𝜏) → 𝜑) → (((𝜏 → 𝜃) → 𝜓) → ((𝜃 ↔ 𝜏) → 𝜒)))
Theorem	bj-bisym 31748	This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.)
⊢ (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → (((𝜓 → 𝜑) → (𝜃 → 𝜒)) → ((𝜑 ↔ 𝜓) → (𝜒 ↔ 𝜃))))
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.
Theorem	bj-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))
Theorem	bj-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⊤)
Theorem	bj-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 ⊢ (∀𝑥𝜑 → 𝜑) (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) → ((∀𝑥𝜑 → 𝜑) → (∀𝑥𝜑 → ∃𝑥𝜑)))
Theorem	bj-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(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥𝜑 ∧ 𝜑)))
Theorem	bj-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 (∀𝑥𝜑 ∧ 𝜑), sometimes called the "strong necessity" of 𝜑. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∀𝑥(∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥𝜑 ∧ 𝜑)) → ∀𝑥(∀𝑥𝜑 ∧ 𝜑)) → (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑))
Theorem	bj-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 → ∀𝑥∀𝑥𝜑) → ((∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥∀𝑥𝜑 → ∀𝑥𝜓)) → (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀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.
Syntax	cprvb 31755	Syntax for the provability predicate.
wff Prv 𝜑
Axiom	ax-prv1 31756	First property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ 𝜑    ⇒   ⊢ Prv 𝜑
Axiom	ax-prv2 31757	Second property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ (Prv (𝜑 → 𝜓) → (Prv 𝜑 → Prv 𝜓))
Axiom	ax-prv3 31758	Third property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ (Prv 𝜑 → Prv Prv 𝜑)
Theorem	prvlem1 31759	An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (Prv 𝜑 → Prv 𝜓)
Theorem	prvlem2 31760	An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (Prv 𝜑 → (Prv 𝜓 → Prv 𝜒))
Theorem	bj-babygodel 31761	See the section header comments for the context. 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 ⊥    ⇒   ⊢ ⊥
Theorem	bj-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 𝜓 → 𝜑))    &   ⊢ (Prv 𝜑 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	bj-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
Syntax	wnff 31764	Syntax for the ℲℲ predicate.
wff ℲℲ𝑥𝜑
Definition	df-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))
Theorem	bj-nf2 31766	Alternate definition of df-bj-nf 31765. (Contributed by BJ, 6-May-2019.)
⊢ (ℲℲ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
Theorem	bj-nf3 31767	Alternate definition of df-bj-nf 31765. (Contributed by BJ, 6-May-2019.)
⊢ (ℲℲ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
Theorem	bj-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))
Theorem	bj-nftht 31769	Closed form of nfth 1718. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥𝜑 → ℲℲ𝑥𝜑)
Theorem	bj-nfntht 31770	Closed form of nfnth 1719. (Contributed by BJ, 6-May-2019.)
⊢ (¬ ∃𝑥𝜑 → ℲℲ𝑥𝜑)
Theorem	bj-nfntht2 31771	Closed form of nfnth 1719. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥 ¬ 𝜑 → ℲℲ𝑥𝜑)
21.14.4.2  Adding ax-gen
Theorem	bj-nfth 31772	Any variable is not free in a theorem. (Contributed by BJ, 6-May-2019.)
⊢ 𝜑    ⇒   ⊢ ℲℲ𝑥𝜑
Theorem	bj-nftru 31773	The true constant has no free variables. (Contributed by BJ, 6-May-2019.)
⊢ ℲℲ𝑥⊤
Theorem	bj-nfnth 31774	Any variable is not free in a falsity. (Contributed by BJ, 6-May-2019.)
⊢ ¬ 𝜑    ⇒   ⊢ ℲℲ𝑥𝜑
Theorem	bj-nffal 31775	The false constant has no free variables. (Contributed by BJ, 6-May-2019.)
⊢ ℲℲ𝑥⊥
Theorem	bj-genr 31776	Generalization rule on the right conjunct. See 19.28 2083. (Contributed by BJ, 7-Jul-2021.)
⊢ (𝜑 ∧ 𝜓)    ⇒   ⊢ (𝜑 ∧ ∀𝑥𝜓)
Theorem	bj-genl 31777	Generalization rule on the left conjunct. See 19.27 2082. (Contributed by BJ, 7-Jul-2021.)
⊢ (𝜑 ∧ 𝜓)    ⇒   ⊢ (∀𝑥𝜑 ∧ 𝜓)
Theorem	bj-genan 31778	Generalization rule on a conjunction. Forward inference associated with 19.26 1786. (Contributed by BJ, 7-Jul-2021.)
⊢ (𝜑 ∧ 𝜓)    ⇒   ⊢ (∀𝑥𝜑 ∧ ∀𝑥𝜓)
21.14.4.3  Adding ax-4
Theorem	bj-2alim 31779	Closed form of 2alimi 1731. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	bj-2exim 31780	Closed form of 2eximi 1753. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓))
Theorem	bj-alanim 31781	Closed form of alanimi 1734. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥((𝜑 ∧ 𝜓) → 𝜒) → ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒))
Theorem	bj-2albi 31782	Closed form of 2albii 1738. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓))
Theorem	bj-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.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ 𝜓)
Theorem	bj-2exbi 31784	Closed form of 2exbii 1765. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓))
Theorem	bj-3exbi 31785	Closed form of 3exbii 1766. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓))
Theorem	bj-sylgt2 31786	Uncurried form of sylgt 1739. (Contributed by BJ, 2-May-2019.)
⊢ ((∀𝑥(𝜓 → 𝜒) ∧ (𝜑 → ∀𝑥𝜓)) → (𝜑 → ∀𝑥𝜒))
Theorem	bj-exlimh 31787	Closed form of close to exlimih 2133. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → 𝜓) → ((∃𝑥𝜓 → 𝜒) → (∃𝑥𝜑 → 𝜒)))
Theorem	bj-exlimh2 31788	Uncurried form of bj-exlimh 31787. (Contributed by BJ, 2-May-2019.)
⊢ ((∀𝑥(𝜑 → 𝜓) ∧ (∃𝑥𝜓 → 𝜒)) → (∃𝑥𝜑 → 𝜒))
Theorem	bj-alrimhi 31789	An inference associated with sylgt 1739 and bj-exlimh 31787. (Contributed by BJ, 12-May-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (ℲℲ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-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)))
Theorem	bj-nexdh2 31791	Uncurried form of bj-nexdh 31790. (Contributed by BJ, 6-May-2019.)
⊢ ((∀𝑥(𝜑 → ¬ 𝜓) ∧ (𝜒 → ∀𝑥𝜑)) → (𝜒 → ¬ ∃𝑥𝜓))
Theorem	bj-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 (∃𝑥𝜑 → 𝜑) instead. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))
Theorem	bj-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(𝜑 ↔ 𝜓) → (ℲℲ𝑥𝜑 ↔ ℲℲ𝑥𝜓))
Theorem	bj-nfxfr 31794	Proof of nfxfr 1771 from bj-nfbi 31793. (Contributed by BJ, 6-May-2019.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ ℲℲ𝑥𝜑    ⇒   ⊢ ℲℲ𝑥𝜓
Theorem	bj-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)
Theorem	bj-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.)
⊢ (∃𝑥𝜓 → 𝜓)    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)
Theorem	bj-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(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥(𝜑 → ¬ 𝜓))
Theorem	bj-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 → (𝜑 → 𝜓))    ⇒   ⊢ (¬ ∀𝑥 ¬ 𝜒 → (∀𝑥𝜑 → ∃𝑥𝜓))
Theorem	bj-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 → (𝜑 → 𝜓))    &   ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)    ⇒   ⊢ (¬ ∀𝑥 ¬ 𝜒 → (∀𝑥𝜑 → 𝜓))
Theorem	bj-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 → (𝜓 → 𝜒))    ⇒   ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))