P. 329 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32801-32900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	knoppndvlem22 32801*	Lemma for knoppndv 32802. (Contributed by Asger C. Ipsen, 19-Aug-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝐷 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐻 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))    ⇒   ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎 ≤ 𝐻 ∧ 𝐻 ≤ 𝑏) ∧ ((𝑏 − 𝑎) < 𝐷 ∧ 𝑎 ≠ 𝑏) ∧ 𝐸 ≤ ((abs‘((𝑊‘𝑏) − (𝑊‘𝑎))) / (𝑏 − 𝑎))))
Theorem	knoppndv 32802*	The continuous nowhere differentiable function 𝑊 ( Knopp, K. (1918). Math. Z. 2, 1-26 ) is, in fact, nowhere differentiable. (Contributed by Asger C. Ipsen, 19-Aug-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))    ⇒   ⊢ (𝜑 → dom (ℝ D 𝑊) = ∅)
Theorem	knoppf 32803*	Knopp's function is a function. (Contributed by Asger C. Ipsen, 25-Aug-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝑊:ℝ⟶ℝ)
Theorem	knoppcn2 32804*	Variant of knoppcn 32771 with different codomain. (Contributed by Asger C. Ipsen, 25-Aug-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))    ⇒   ⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℝ))
Theorem	cnndvlem1 32805*	Lemma for cnndv 32807. (Contributed by Asger C. Ipsen, 25-Aug-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ (((1 / 2)↑𝑛) · (𝑇‘(((2 · 3)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    ⇒   ⊢ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅)
Theorem	cnndvlem2 32806*	Lemma for cnndv 32807. (Contributed by Asger C. Ipsen, 26-Aug-2021.)
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ (((1 / 2)↑𝑛) · (𝑇‘(((2 · 3)↑𝑛) · 𝑦)))))    &   ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖))    ⇒   ⊢ ∃𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅)
Theorem	cnndv 32807	There exists a continuous nowhere differentiable function. The result follows directly from knoppcn 32771 and knoppndv 32802. (Contributed by Asger C. Ipsen, 26-Aug-2021.)
⊢ ∃𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅)
20.14  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.
20.14.1  Propositional calculus Miscellaneous utility theorems of propositional calculus.
20.14.1.1  Derived rules of inference In this section, we prove a few rules of inference derived from modus ponens ax-mp 5, and which do not depend on any other axioms.
Theorem	bj-mp2c 32808	A double modus ponens inference. Inference associated with mpd 15. (Contributed by BJ, 24-Sep-2019.)
⊢ 𝜑    &   ⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ 𝜒
Theorem	bj-mp2d 32809	A double modus ponens inference. Inference associated with mpcom 38. (Contributed by BJ, 24-Sep-2019.)
⊢ 𝜑    &   ⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 → (𝜑 → 𝜒))    ⇒   ⊢ 𝜒
20.14.1.2  A syntactic theorem In this section, we prove a syntactic theorem (bj-0 32810) asserting that some formula is well-formed. Then, we use this syntactic theorem to shorten the proof of a "usual" theorem (bj-1 32811) and explain in the comment of that theorem why this phenomenon is unusual.
Theorem	bj-0 32810	A syntactic theorem. See the section comment and the comment of bj-1 32811. 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 2128 and weq 2028. (Contributed by BJ, 24-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
wff ((𝜑 → 𝜓) → 𝜒)
Theorem	bj-1 32811	In this proof, the use of the syntactic theorem bj-0 32810 allows to reduce the total length by one (non-essential) step. See also the section comment and the comment of bj-0 32810. Since bj-0 32810 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 32810 as a theorem referencing it). The full proof reads $= wph wps wch bj-0 id $. (while, without using bj-0 32810, 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 2028 or wel 2128). 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 32811 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))
20.14.1.3  Minimal implicational calculus
Theorem	bj-a1k 32812	Weakening of ax-1 6. This shortens the proofs of dfwe2 7134 (937>925), ordunisuc2 7197 (789>777), r111 8799 (558>545), smo11 7618 (1176>1164). (Contributed by BJ, 11-Aug-2020.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜓)))
Theorem	bj-jarri 32813	Inference associated with jarr 106. Its associated inference is bj-jarrii 32814. (Contributed by BJ, 29-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜒)    ⇒   ⊢ (𝜓 → 𝜒)
Theorem	bj-jarrii 32814	Inference associated with bj-jarri 32813. (Contributed by BJ, 29-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜒)    &   ⊢ 𝜓    ⇒   ⊢ 𝜒
Theorem	bj-imim2ALT 32815	More direct proof of imim2 58. Note that imim2i 16 and imim2d 57 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 32816	The propositional function (𝜒 → (. → 𝜃)) is decreasing. (Contributed by BJ, 19-Jul-2019.)
⊢ ((𝜑 → 𝜓) → ((𝜒 → (𝜓 → 𝜃)) → (𝜒 → (𝜑 → 𝜃))))
Theorem	bj-imim21i 32817	Inference associated with bj-imim21 32816. Its associated inference is syl5 34. (Contributed by BJ, 19-Jul-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜒 → (𝜓 → 𝜃)) → (𝜒 → (𝜑 → 𝜃)))
20.14.1.4  Positive calculus
Theorem	bj-orim2 32818	Proof of orim2 922 from the axiomatic definition of disjunction (olc 398, orc 399, jao 535) and minimal implicational calculus. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜒 ∨ 𝜑) → (𝜒 ∨ 𝜓)))
Theorem	bj-curry 32819	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 32820	Proof of peirce 193 from minimal implicational calculus, the axiomatic definition of disjunction (olc 398, orc 399, jao 535), and Curry's axiom bj-curry 32819. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
Theorem	bj-currypeirce 32821	Curry's axiom (a non-intuitionistic statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 193 over minimal implicational calculus and the axiomatic definition of disjunction (olc 398, orc 399, jao 535). A shorter proof from bj-orim2 32818, pm1.2 536, syl6com 37 is possible if we accept to use pm1.2 536, itself a direct consequence of jao 535. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∨ (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
Theorem	bj-peircecurry 32822	Peirce's axiom peirce 193 implies Curry's axiom over minimal implicational calculus and the axiomatic definition of disjunction (olc 398, orc 399, jao 535). See comment of bj-currypeirce 32821. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 ∨ (𝜑 → 𝜓))
20.14.1.5  Implication and negation
Theorem	pm4.81ALT 32823	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 32824	Alternate proof of con4i 113. 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 32825	A commuted form of the contrapositive, true in minimal calculus. (Contributed by BJ, 19-Mar-2020.)
⊢ (𝜑 → ((𝜓 → ¬ 𝜑) → ¬ 𝜓))
Theorem	bj-con2comi 32826	Inference associated with bj-con2com 32825. Its associated inference is mt2 191. TODO: when in the main part, add to mt2 191 that it is the inference associated with bj-con2comi 32826. (Contributed by BJ, 19-Mar-2020.)
⊢ 𝜑    ⇒   ⊢ ((𝜓 → ¬ 𝜑) → ¬ 𝜓)
Theorem	bj-pm2.01i 32827	Inference associated with pm2.01 180. (Contributed by BJ, 30-Mar-2020.)
⊢ (𝜑 → ¬ 𝜑)    ⇒   ⊢ ¬ 𝜑
Theorem	bj-nimn 32828	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 159, however, the present proof uses theorems that are more basic than jc 159. (Proof modification is discouraged.)
⊢ (𝜑 → ¬ (𝜑 → ¬ 𝜑))
Theorem	bj-nimni 32829	Inference associated with bj-nimn 32828. (Contributed by BJ, 19-Mar-2020.)
⊢ 𝜑    ⇒   ⊢ ¬ (𝜑 → ¬ 𝜑)
Theorem	bj-peircei 32830	Inference associated with peirce 193. (Contributed by BJ, 30-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜑)    ⇒   ⊢ 𝜑
Theorem	bj-looinvi 32831	Inference associated with looinv 194. Its associated inference is bj-looinvii 32832. (Contributed by BJ, 30-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜓)    ⇒   ⊢ ((𝜓 → 𝜑) → 𝜑)
Theorem	bj-looinvii 32832	Inference associated with bj-looinvi 32831. (Contributed by BJ, 30-Mar-2020.)
⊢ ((𝜑 → 𝜓) → 𝜓)    &   ⊢ (𝜓 → 𝜑)    ⇒   ⊢ 𝜑
20.14.1.6  Disjunction A few lemmas about disjunction. The fundamental theorems in this family are the dual statements pm4.71 665 and pm4.72 956. See also biort 974 and biorf 419.
Theorem	bj-jaoi1 32833	Shortens orfa2 34169 (58>53), pm1.2 536 (20>18), pm1.2 536 (20>18), pm2.4 600 (31>25), pm2.41 598 (31>25), pm2.42 599 (38>32), pm3.2ni 935 (43>39), pm4.44 602 (55>51). (Contributed by BJ, 30-Sep-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜑 ∨ 𝜓) → 𝜓)
Theorem	bj-jaoi2 32834	Shortens consensus 1041 (110>106), elnn0z 11553 (336>329), pm1.2 536 (20>19), pm3.2ni 935 (43>39), pm4.44 602 (55>51). (Contributed by BJ, 30-Sep-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜓 ∨ 𝜑) → 𝜓)
20.14.1.7  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 510 through pm4.57 519.
Theorem	bj-dfbi4 32835	Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓)))
Theorem	bj-dfbi5 32836	Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)))
Theorem	bj-dfbi6 32837	Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∧ 𝜓)))
Theorem	bj-bijust0 32838	The general statement that bijust 195 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))
20.14.1.8  The conditional operator for propositions
Theorem	bj-consensus 32839	Version of consensus 1041 expressed using the conditional operator. (Remark: it may be better to express it as consensus 1041, 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 32840	Alternate proof of bj-consensus 32839. (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 32841*	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 32842*	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 2735. (Contributed by BJ, 20-Sep-2019.)
⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}
Theorem	bj-ififc 32843*	A theorem linking if- and if. (Contributed by BJ, 24-Sep-2019.)
⊢ (𝑥 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵))
20.14.1.9  Propositional calculus: miscellaneous Miscellaneous theorems of propositional calculus.
Theorem	bj-imbi12 32844	Uncurried (imported) form of imbi12 335. (Contributed by BJ, 6-May-2019.)
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))
Theorem	bj-biorfi 32845	This should be labeled "biorfi" while the current biorfi 421 should be labeled "biorfri". The dual of biorf 419 is not biantr 1010 but iba 525 (and ibar 526). 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 32846	Dual of truan 1638 (which has biconditional reversed). (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
⊢ (𝜑 ↔ (⊥ ∨ 𝜑))
Theorem	bj-falor2 32847	Dual of truan 1638. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
⊢ ((⊥ ∨ 𝜑) ↔ 𝜑)
Theorem	bj-bibibi 32848	A property of the biconditional. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
⊢ (𝜑 ↔ (𝜓 ↔ (𝜑 ↔ 𝜓)))
Theorem	bj-imn3ani 32849	Duplication of bnj1224 31150. 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 32850	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 32851	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 32852	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))))
20.14.2  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 1859 corresponds to the necessitation rule of modal logic, and ax-4 1874 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 32857.
Theorem	bj-axdd2 32853	This implication, proved using only ax-gen 1859 and ax-4 1874 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 32854. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))
Theorem	bj-axd2d 32854	This implication, proved using only ax-gen 1859 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 32853. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∀𝑥⊤ → ∃𝑥⊤) → ∃𝑥⊤)
Theorem	bj-axtd 32855	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 32853 and bj-axd2d 32854. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → ((∀𝑥𝜑 → 𝜑) → (∀𝑥𝜑 → ∃𝑥𝜑)))
Theorem	bj-gl4lem 32856	Lemma for bj-gl4 32857. 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 32857	In a normal modal logic, the modal axiom GL implies the modal axiom (4). Note that the antecedent of bj-gl4 32857 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 32858	Over minimal calculus, the modal axiom (4) (hba1 2286) and the modal axiom (K) (ax-4 1874) together imply axc4 2265. (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))))
20.14.3  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 32860 and ax-prv2 32861 and ax-prv3 32862. Note the similarity with ax-gen 1859, ax-4 1874 and hba1 2286 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 32865) and Löb's theorem (bj-babylob 32866). See the comments of these theorems for details.
Syntax	cprvb 32859	Syntax for the provability predicate.
wff Prv 𝜑
Axiom	ax-prv1 32860	First property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ 𝜑    ⇒   ⊢ Prv 𝜑
Axiom	ax-prv2 32861	Second property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ (Prv (𝜑 → 𝜓) → (Prv 𝜑 → Prv 𝜓))
Axiom	ax-prv3 32862	Third property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ (Prv 𝜑 → Prv Prv 𝜑)
Theorem	prvlem1 32863	An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (Prv 𝜑 → Prv 𝜓)
Theorem	prvlem2 32864	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 32865	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 32866	See the section header comments for the context, as well as the comments for bj-babygodel 32865. 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 32867	Proof of Gödel's theorem from Löb's theorem (see comments at bj-babygodel 32865 and bj-babylob 32866 for details). (Contributed by BJ, 20-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 ↔ ¬ Prv 𝜑)    &   ⊢ ¬ Prv ⊥    ⇒   ⊢ ⊥
20.14.4  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.
20.14.4.1  Adding ax-gen
Theorem	bj-genr 32868	Generalization rule on the right conjunct. See 19.28 2231. (Contributed by BJ, 7-Jul-2021.)
⊢ (𝜑 ∧ 𝜓)    ⇒   ⊢ (𝜑 ∧ ∀𝑥𝜓)
Theorem	bj-genl 32869	Generalization rule on the left conjunct. See 19.27 2230. (Contributed by BJ, 7-Jul-2021.)
⊢ (𝜑 ∧ 𝜓)    ⇒   ⊢ (∀𝑥𝜑 ∧ 𝜓)
Theorem	bj-genan 32870	Generalization rule on a conjunction. Forward inference associated with 19.26 1935. (Contributed by BJ, 7-Jul-2021.)
⊢ (𝜑 ∧ 𝜓)    ⇒   ⊢ (∀𝑥𝜑 ∧ ∀𝑥𝜓)
20.14.4.2  Adding ax-4
Theorem	bj-2alim 32871	Closed form of 2alimi 1877. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	bj-2exim 32872	Closed form of 2eximi 1900. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓))
Theorem	bj-alanim 32873	Closed form of alanimi 1881. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥((𝜑 ∧ 𝜓) → 𝜒) → ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒))
Theorem	bj-2albi 32874	Closed form of 2albii 1885. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓))
Theorem	bj-notalbii 32875	Equivalence of universal quantification of negation of equivalent formulas. Shortens ab0 4082 (103>94), ballotlem2 30830 (2655>2648), bnj1143 31139 (522>519), hausdiag 21621 (2119>2104). (Contributed by BJ, 17-Jul-2021.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ 𝜓)
Theorem	bj-2exbi 32876	Closed form of 2exbii 1912. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓))
Theorem	bj-3exbi 32877	Closed form of 3exbii 1913. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓))
Theorem	bj-sylgt2 32878	Uncurried (imported) form of sylgt 1886. (Contributed by BJ, 2-May-2019.)
⊢ ((∀𝑥(𝜓 → 𝜒) ∧ (𝜑 → ∀𝑥𝜓)) → (𝜑 → ∀𝑥𝜒))
Theorem	bj-exlimh 32879	Closed form of close to exlimih 2283. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜑 → 𝜓) → ((∃𝑥𝜓 → 𝜒) → (∃𝑥𝜑 → 𝜒)))
Theorem	bj-exlimh2 32880	Uncurried (imported) form of bj-exlimh 32879. (Contributed by BJ, 2-May-2019.)
⊢ ((∀𝑥(𝜑 → 𝜓) ∧ (∃𝑥𝜓 → 𝜒)) → (∃𝑥𝜑 → 𝜒))
Theorem	bj-alrimhi 32881	An inference associated with sylgt 1886 and bj-exlimh 32879. (Contributed by BJ, 12-May-2019.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜓))
Theorem	bj-alexim 32882	Closed form of aleximi 1896 (with a shorter proof, so aleximi 1896 could be deduced from it -- exim 1898 would have to be proved first, but its proof is shorter (currently almost a subproof of aleximi 1896)). (Contributed by BJ, 8-Nov-2021.)
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)))
Theorem	bj-nexdh 32883	Closed form of nexdh 1929 (actually, its general instance). (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓)))
Theorem	bj-nexdh2 32884	Uncurried (imported) form of bj-nexdh 32883. (Contributed by BJ, 6-May-2019.)
⊢ ((∀𝑥(𝜑 → ¬ 𝜓) ∧ (𝜒 → ∀𝑥𝜑)) → (𝜒 → ¬ ∃𝑥𝜓))
Theorem	bj-hbxfrbi 32885	Closed form of hbxfrbi 1889. Notes: it is less important than nfbiit 1914; it requires sp 2188 (unlike nfbiit 1914); there is an obvious version with (∃𝑥𝜑 → 𝜑) instead. (Contributed by BJ, 6-May-2019.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))
Theorem	bj-exlime 32886	Variant of exlimih 2283 where the non-freeness of 𝑥 in 𝜓 is expressed using an existential quantifier, thus requiring fewer axioms. (Contributed by BJ, 17-Mar-2020.)
⊢ (∃𝑥𝜓 → 𝜓)    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)
Theorem	bj-exnalimn 32887	A transformation of quantifiers and logical connectives. The general statement that equs3 2029 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 1976. I propose to move to the main part: bj-exnalimn 32887, bj-exalim 32888, bj-exalimi 32889, bj-exalims 32890, bj-exalimsi 32891, bj-ax12i 32893, bj-ax12wlem 32894, bj-ax12w 32942, and remove equs3 2029. A new label is needed for bj-ax12i 32893 and label suggestions are welcome for the others. I also propose to change ¬ ∀𝑥¬ to ∃𝑥 in speimfw 2030 and spimfw 2032 (other spim* theorems use ∃𝑥 and very few theorems in set.mm use ¬ ∀𝑥¬). (Contributed by BJ, 29-Sep-2019.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥(𝜑 → ¬ 𝜓))
Theorem	bj-exalim 32888	Distributing quantifiers over a double implication. (Contributed by BJ, 8-Nov-2021.)
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)))
Theorem	bj-exalimi 32889	An inference for distributing quantifiers over a double implication. (Almost) the general statement that speimfw 2030 proves. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))
Theorem	bj-exalims 32890	Distributing quantifiers over a double implication. (Almost) the general statement that spimfw 2032 proves. (Contributed by BJ, 29-Sep-2019.)
⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))    ⇒   ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒)))
Theorem	bj-exalimsi 32891	An inference for distributing quantifiers over a double implication. (Almost) the general statement that spimfw 2032 proves. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))    ⇒   ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	bj-ax12ig 32892	A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 32893. (Contributed by BJ, 19-Dec-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
Theorem	bj-ax12i 32893	A weakening of bj-ax12ig 32892 that is sufficient to prove a weak form of the axiom of substitution ax-12 2184. The general statement of which ax12i 2033 is an instance. (Contributed by BJ, 29-Sep-2019.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜒 → ∀𝑥𝜒)    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
20.14.4.3  Adding ax-5
Theorem	bj-ax12wlem 32894*	A lemma used to prove a weak version of the axiom of substitution ax-12 2184. (Temporary comment: The general statement that ax12wlem 2146 proves.) (Contributed by BJ, 20-Mar-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 → 𝜓)))
20.14.4.4  Equality and substitution
Theorem	bj-ssbjust 32895*	Justification theorem for df-ssb 32897 from Tarski's FOL. (Contributed by BJ, 9-Nov-2021.)
⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
Syntax	wssb 32896	Syntax for the substitution of a variable for a variable in a formula. (Contributed by BJ, 22-Dec-2020.)
wff [𝑡/𝑥]b𝜑
Definition	df-ssb 32897*	Alternate definition of proper substitution. Note that the occurrences of a given variable are either all bound (𝑥, 𝑦) or all free (𝑡). Also note that the definiens uses only primitive symbols. It is obtained by applying twice Tarski's definition sb6 2554 which is valid for disjoint variables, so we introduce a dummy variable 𝑦 to isolate 𝑥 from 𝑡, as in dfsb7 2580 with respect to sb5 2555. This double level definition will make several proofs using it appear as doubled. Alternately, one could often first prove as a lemma the same theorem with a DV condition on the substitute and the substituted variables, and then prove the original theorem by applying this lemma twice in a row. This definition uses a dummy variable, but the justification theorem, bj-ssbjust 32895, is provable from Tarski's FOL. Once this is proved, more of the fundamental properties of proper substitution will be provable from Tarski's FOL system, sometimes with the help of specialization sp 2188, of the substitution axiom ax-12 2184, and of commutation of quantifiers ax-11 2171; that is, ax-13 2379 will often be avoided. (Contributed by BJ, 22-Dec-2020.)
⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	bj-ssbim 32898	Distribute substitution over implication, closed form. Specialization of implication. Uses only ax-1--5. Compare spsbim 2519. (Contributed by BJ, 22-Dec-2020.)
⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓))
Theorem	bj-ssbbi 32899	Biconditional property for substitution, closed form. Specialization of biconditional. Uses only ax-1--5. Compare spsbbi 2527. (Contributed by BJ, 22-Dec-2020.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓))
Theorem	bj-ssbimi 32900	Distribute substitution over implication. Uses only ax-1--5. (Contributed by BJ, 22-Dec-2020.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓)