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Theorem List for Metamath Proof Explorer - 33101-33200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremknoppndvlem13 33101 Lemma for knoppndv 33111. (Contributed by Asger C. Ipsen, 1-Jul-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
(𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑𝐶 ≠ 0)

Theoremknoppndvlem14 33102* Lemma for knoppndv 33111. (Contributed by Asger C. Ipsen, 1-Jul-2021.) (Revised by Asger C. Ipsen, 7-Jul-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹𝐴)‘𝑖))) ≤ ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))))

Theoremknoppndvlem15 33103* Lemma for knoppndv 33111. (Contributed by Asger C. Ipsen, 6-Jul-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ (abs‘((𝑊𝐵) − (𝑊𝐴))))

Theoremknoppndvlem16 33104 Lemma for knoppndv 33111. (Contributed by Asger C. Ipsen, 19-Jul-2021.)
𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝐵𝐴) = (((2 · 𝑁)↑-𝐽) / 2))

Theoremknoppndvlem17 33105* Lemma for knoppndv 33111. (Contributed by Asger C. Ipsen, 12-Aug-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → ((((2 · 𝑁) · (abs‘𝐶))↑𝐽) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ ((abs‘((𝑊𝐵) − (𝑊𝐴))) / (𝐵𝐴)))

Theoremknoppndvlem18 33106* Lemma for knoppndv 33111. (Contributed by Asger C. Ipsen, 14-Aug-2021.)
(𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐺 ∈ ℝ+)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → ∃𝑗 ∈ ℕ0 ((((2 · 𝑁)↑-𝑗) / 2) < 𝐷𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · 𝐺)))

Theoremknoppndvlem19 33107* Lemma for knoppndv 33111. (Contributed by Asger C. Ipsen, 17-Aug-2021.)
𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑚)    &   𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑚 + 1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝐻 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → ∃𝑚 ∈ ℤ (𝐴𝐻𝐻𝐵))

Theoremknoppndvlem20 33108 Lemma for knoppndv 33111. (Contributed by Asger C. Ipsen, 18-Aug-2021.)
(𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ+)

Theoremknoppndvlem21 33109* Lemma for knoppndv 33111. (Contributed by Asger C. Ipsen, 18-Aug-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   𝐺 = (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐻 ∈ ℝ)    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))    &   (𝜑 → (((2 · 𝑁)↑-𝐽) / 2) < 𝐷)    &   (𝜑𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝐽) · 𝐺))       (𝜑 → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎𝐻𝐻𝑏) ∧ ((𝑏𝑎) < 𝐷𝑎𝑏) ∧ 𝐸 ≤ ((abs‘((𝑊𝑏) − (𝑊𝑎))) / (𝑏𝑎))))

Theoremknoppndvlem22 33110* Lemma for knoppndv 33111. (Contributed by Asger C. Ipsen, 19-Aug-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐻 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎𝐻𝐻𝑏) ∧ ((𝑏𝑎) < 𝐷𝑎𝑏) ∧ 𝐸 ≤ ((abs‘((𝑊𝑏) − (𝑊𝑎))) / (𝑏𝑎))))

Theoremknoppndv 33111* 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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → dom (ℝ D 𝑊) = ∅)

Theoremknoppf 33112* 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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)       (𝜑𝑊:ℝ⟶ℝ)

Theoremknoppcn2 33113* Variant of knoppcn 33081 with different codomain. (Contributed by Asger C. Ipsen, 25-Aug-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ (-1(,)1))       (𝜑𝑊 ∈ (ℝ–cn→ℝ))

Theoremcnndvlem1 33114* Lemma for cnndv 33116. (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 ((𝐹𝑤)‘𝑖))       (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅)

Theoremcnndvlem2 33115* Lemma for cnndv 33116. (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 ((𝐹𝑤)‘𝑖))       𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅)

Theoremcnndv 33116 There exists a continuous nowhere differentiable function. The result follows directly from knoppcn 33081 and knoppndv 33111. (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.

Theorembj-mp2c 33117 A double modus ponens inference. Inference associated with mpd 15. (Contributed by BJ, 24-Sep-2019.)
𝜑    &   (𝜑𝜓)    &   (𝜑 → (𝜓𝜒))       𝜒

Theorembj-mp2d 33118 A double modus ponens inference. Inference associated with mpcom 38. (Contributed by BJ, 24-Sep-2019.)
𝜑    &   (𝜑𝜓)    &   (𝜓 → (𝜑𝜒))       𝜒

20.14.1.2  A syntactic theorem

In this section, we prove a syntactic theorem (bj-0 33119) asserting that some formula is well-formed. Then, we use this syntactic theorem to shorten the proof of a "usual" theorem (bj-1 33120) and explain in the comment of that theorem why this phenomenon is unusual.

Theorembj-0 33119 A syntactic theorem. See the section comment and the comment of bj-1 33120. 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 2108 and weq 2005. (Contributed by BJ, 24-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
wff ((𝜑𝜓) → 𝜒)

Theorembj-1 33120 In this proof, the use of the syntactic theorem bj-0 33119 allows to reduce the total length by one (non-essential) step. See also the section comment and the comment of bj-0 33119. Since bj-0 33119 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 33119 as a theorem referencing it). The full proof reads \$= wph wps wch bj-0 id \$. (while, without using bj-0 33119, 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 2005 or wel 2108). 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 33120 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))

20.14.1.3  Minimal implicational calculus

Theorembj-a1k 33121 Weakening of ax-1 6. This shortens the proofs of dfwe2 7261 (937>925), ordunisuc2 7324 (789>777), r111 8937 (558>545), smo11 7746 (1176>1164). (Contributed by BJ, 11-Aug-2020.)
(𝜑 → (𝜓 → (𝜒𝜓)))

Theorembj-jarrii 33122 Inference associated with jarri 107. (Contributed by BJ, 29-Mar-2020.)
((𝜑𝜓) → 𝜒)    &   𝜓       𝜒

Theorembj-imim21 33123 The propositional function (𝜒 → (. → 𝜃)) is decreasing. (Contributed by BJ, 19-Jul-2019.)
((𝜑𝜓) → ((𝜒 → (𝜓𝜃)) → (𝜒 → (𝜑𝜃))))

Theorembj-imim21i 33124 Inference associated with bj-imim21 33123. Its associated inference is syl5 34. (Contributed by BJ, 19-Jul-2019.)
(𝜑𝜓)       ((𝜒 → (𝜓𝜃)) → (𝜒 → (𝜑𝜃)))

20.14.1.4  Positive calculus

Theorembj-syl66ib 33125 A mixed syllogism inference derived from syl6ib 243. In addition to bj-dvelimdv1 33413, it can also shorten alexsubALTlem4 22266 (4821>4812), supsrlem 10270 (2868>2863). (Contributed by BJ, 20-Oct-2021.)
(𝜑 → (𝜓𝜃))    &   (𝜃𝜏)    &   (𝜏𝜒)       (𝜑 → (𝜓𝜒))

Theorembj-orim2 33126 Proof of orim2 953 from the axiomatic definition of disjunction (olc 857, orc 856, jao 946) and minimal implicational calculus. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))

Theorembj-peirce 33127 Proof of peirce 194 from minimal implicational calculus, the axiomatic definition of disjunction (olc 857, orc 856, jao 946), and Curry's axiom curryax 880. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜑) → 𝜑)

Theorembj-currypeirce 33128 Curry's axiom (a non-intuitionistic statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 194 over minimal implicational calculus and the axiomatic definition of disjunction (olc 857, orc 856, jao 946). A shorter proof from bj-orim2 33126, pm1.2 890, syl6com 37 is possible if we accept to use pm1.2 890, itself a direct consequence of jao 946. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∨ (𝜑𝜓)) → (((𝜑𝜓) → 𝜑) → 𝜑))

Theorembj-peircecurry 33129 Peirce's axiom peirce 194 implies Curry's axiom over minimal implicational calculus and the axiomatic definition of disjunction (olc 857, orc 856, jao 946). See comment of bj-currypeirce 33128. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ∨ (𝜑𝜓))

20.14.1.5  Implication and negation

Theorembj-con2com 33130 A commuted form of the contrapositive, true in minimal calculus. (Contributed by BJ, 19-Mar-2020.)
(𝜑 → ((𝜓 → ¬ 𝜑) → ¬ 𝜓))

Theorembj-con2comi 33131 Inference associated with bj-con2com 33130. Its associated inference is mt2 192. TODO: when in the main part, add to mt2 192 that it is the inference associated with bj-con2comi 33131. (Contributed by BJ, 19-Mar-2020.)
𝜑       ((𝜓 → ¬ 𝜑) → ¬ 𝜓)

Theorembj-pm2.01i 33132 Inference associated with the weak Clavius law pm2.01 181. (Contributed by BJ, 30-Mar-2020.)
(𝜑 → ¬ 𝜑)        ¬ 𝜑

Theorembj-nimn 33133 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 161, however, the present proof uses theorems that are more basic than jc 161. (Proof modification is discouraged.)
(𝜑 → ¬ (𝜑 → ¬ 𝜑))

Theorembj-nimni 33134 Inference associated with bj-nimn 33133. (Contributed by BJ, 19-Mar-2020.)
𝜑        ¬ (𝜑 → ¬ 𝜑)

Theorembj-peircei 33135 Inference associated with peirce 194. (Contributed by BJ, 30-Mar-2020.)
((𝜑𝜓) → 𝜑)       𝜑

Theorembj-looinvi 33136 Inference associated with looinv 195. Its associated inference is bj-looinvii 33137. (Contributed by BJ, 30-Mar-2020.)
((𝜑𝜓) → 𝜓)       ((𝜓𝜑) → 𝜑)

Theorembj-looinvii 33137 Inference associated with bj-looinvi 33136. (Contributed by BJ, 30-Mar-2020.)
((𝜑𝜓) → 𝜓)    &   (𝜓𝜑)       𝜑

20.14.1.6  Disjunction

A few lemmas about disjunction. The fundamental theorems in this family are the dual statements pm4.71 553 and pm4.72 935. See also biort 922 and biorf 923.

Theorembj-jaoi1 33138 Shortens orfa2 34516 (58>53), pm1.2 890 (20>18), pm1.2 890 (20>18), pm2.4 893 (31>25), pm2.41 894 (31>25), pm2.42 929 (38>32), pm3.2ni 867 (43>39), pm4.44 982 (55>51). (Contributed by BJ, 30-Sep-2019.)
(𝜑𝜓)       ((𝜑𝜓) → 𝜓)

Theorembj-jaoi2 33139 Shortens consensus 1036 (110>106), elnn0z 11745 (336>329), pm1.2 890 (20>19), pm3.2ni 867 (43>39), pm4.44 982 (55>51). (Contributed by BJ, 30-Sep-2019.)
(𝜑𝜓)       ((𝜓𝜑) → 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 837, df-an 387, pm4.64 838, imor 842, pm4.62 845 through pm4.67 389, and, for the De Morgan laws, ianor 967 through pm4.57 976.

Theorembj-dfbi4 33140 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Theorembj-dfbi5 33141 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) ↔ ((𝜑𝜓) → (𝜑𝜓)))

Theorembj-dfbi6 33142 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))

Theorembj-bijust0ALT 33143 Alternate proof of bijust0 196; shorter but using additional intermediate results. (Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Revised by BJ, 19-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ((𝜑𝜑) → ¬ (𝜑𝜑))

Theorembj-bijust00 33144 A self-implication does not imply the negation of a self-implication. Most general theorem of which bijust 197 is an instance (bijust0 196 and bj-bijust0ALT 33143 are therefore also instances of it). (Contributed by BJ, 7-Sep-2022.)
¬ ((𝜑𝜑) → ¬ (𝜓𝜓))

20.14.1.8  The conditional operator for propositions

Theorembj-consensus 33145 Version of consensus 1036 expressed using the conditional operator. (Remark: it may be better to express it as consensus 1036, 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 33146 Alternate proof of bj-consensus 33145. (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 33147* 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 33148* 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 2764. (Contributed by BJ, 20-Sep-2019.)
if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)}

Theorembj-ififc 33149* A theorem linking if- and if. (Contributed by BJ, 24-Sep-2019.)
(𝑥 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑥𝐴, 𝑥𝐵))

20.14.1.9  Propositional calculus: miscellaneous

Miscellaneous theorems of propositional calculus.

Theorembj-imbi12 33150 Uncurried (imported) form of imbi12 338. (Contributed by BJ, 6-May-2019.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) ↔ (𝜓𝜃)))

Theorembj-biorfi 33151 This should be labeled "biorfi" while the current biorfi 925 should be labeled "biorfri". The dual of biorf 923 is not biantr 796 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 33152 Dual of truan 1613 (which has biconditional reversed). (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
(𝜑 ↔ (⊥ ∨ 𝜑))

Theorembj-falor2 33153 Dual of truan 1613. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
((⊥ ∨ 𝜑) ↔ 𝜑)

Theorembj-bibibi 33154 A property of the biconditional. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
(𝜑 ↔ (𝜓 ↔ (𝜑𝜓)))

Theorembj-imn3ani 33155 Duplication of bnj1224 31475. Three-fold version of imnani 391. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by BJ, 22-Oct-2019.) (Proof modification is discouraged.)
¬ (𝜑𝜓𝜒)       ((𝜑𝜓) → ¬ 𝜒)

Theorembj-andnotim 33156 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 33157 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 33158 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))))

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 disjoint variable conditions between 𝑥 and any other metavariables appearing in the statements.)

For instance, ax-gen 1839 corresponds to the necessitation rule of modal logic, and ax-4 1853 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 33163.

Theorembj-axdd2 33159 This implication, proved using only ax-gen 1839 and ax-4 1853 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 33160. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))

Theorembj-axd2d 33160 This implication, proved using only ax-gen 1839 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 33159. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥⊤ → ∃𝑥⊤) → ∃𝑥⊤)

Theorembj-axtd 33161 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 33159 and bj-axd2d 33160. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → ((∀𝑥𝜑𝜑) → (∀𝑥𝜑 → ∃𝑥𝜑)))

Theorembj-gl4lem 33162 Lemma for bj-gl4 33163. 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 33163 In a normal modal logic, the modal axiom GL implies the modal axiom (4). Note that the antecedent of bj-gl4 33163 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 33164 Over minimal calculus, the modal axiom (4) (hba1 2268) and the modal axiom (K) (ax-4 1853) together imply axc4 2296. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥𝜑 → ∀𝑥𝑥𝜑) → ((∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝑥𝜑 → ∀𝑥𝜓)) → (∀𝑥(∀𝑥𝜑𝜓) → (∀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 33166 and ax-prv2 33167 and ax-prv3 33168. Note the similarity with ax-gen 1839, ax-4 1853 and hba1 2268 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 33171) and Löb's theorem (bj-babylob 33172). See the comments of these theorems for details.

Syntaxcprvb 33165 Syntax for the provability predicate.
wff Prv 𝜑

Axiomax-prv1 33166 First property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
𝜑       Prv 𝜑

Axiomax-prv2 33167 Second property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(Prv (𝜑𝜓) → (Prv 𝜑 → Prv 𝜓))

Axiomax-prv3 33168 Third property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(Prv 𝜑 → Prv Prv 𝜑)

Theoremprvlem1 33169 An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(𝜑𝜓)       (Prv 𝜑 → Prv 𝜓)

Theoremprvlem2 33170 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 33172 See the section header comments for the context, as well as the comments for bj-babygodel 33171.

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 33173 Proof of Gödel's theorem from Löb's theorem (see comments at bj-babygodel 33171 and bj-babylob 33172 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 disjoint variable conditions, or disjoint variable conditions replaced with non-freeness hypotheses...). Sorted in the same order as in the main part.

Theorembj-genr 33174 Generalization rule on the right conjunct. See 19.28 2214. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (𝜑 ∧ ∀𝑥𝜓)

Theorembj-genl 33175 Generalization rule on the left conjunct. See 19.27 2213. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theorembj-genan 33176 Generalization rule on a conjunction. Forward inference associated with 19.26 1916. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (∀𝑥𝜑 ∧ ∀𝑥𝜓)

Theorembj-2alim 33177 Closed form of 2alimi 1856. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∀𝑥𝑦𝜑 → ∀𝑥𝑦𝜓))

Theorembj-2exim 33178 Closed form of 2eximi 1879. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 → ∃𝑥𝑦𝜓))

Theorembj-alanim 33179 Closed form of alanimi 1860. (Contributed by BJ, 6-May-2019.)
(∀𝑥((𝜑𝜓) → 𝜒) → ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒))

Theorembj-2albi 33180 Closed form of 2albii 1864. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦𝜓))

Theorembj-notalbii 33181 Equivalence of universal quantification of negation of equivalent formulas. Shortens ab0 4182 (103>94), ballotlem2 31153 (2655>2648), bnj1143 31464 (522>519), hausdiag 21861 (2119>2104). (Contributed by BJ, 17-Jul-2021.)
(𝜑𝜓)       (∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ 𝜓)

Theorembj-2exbi 33182 Closed form of 2exbii 1893. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝜓))

Theorembj-3exbi 33183 Closed form of 3exbii 1894. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦𝑧(𝜑𝜓) → (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓))

Theorembj-sylgt2 33184 Uncurried (imported) form of sylgt 1865. (Contributed by BJ, 2-May-2019.)
((∀𝑥(𝜓𝜒) ∧ (𝜑 → ∀𝑥𝜓)) → (𝜑 → ∀𝑥𝜒))

Theorembj-exlimh 33185 Closed form of close to exlimih 2263. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑𝜓) → ((∃𝑥𝜓𝜒) → (∃𝑥𝜑𝜒)))

Theorembj-exlimh2 33186 Uncurried (imported) form of bj-exlimh 33185. (Contributed by BJ, 2-May-2019.)
((∀𝑥(𝜑𝜓) ∧ (∃𝑥𝜓𝜒)) → (∃𝑥𝜑𝜒))

Theorembj-exlalrim 33187 Adding an existential quantifier in the antecedent and a universal quantifier in the consequent of an implication. Note that the half of the proof using the antecedent 𝑥𝜑 does not require ax-gen 1839. See also 19.38a 1883 and 19.38b 1885. (Contributed by BJ, 27-Nov-2022.)
((Ⅎ𝑥𝜑 ∨ Ⅎ𝑥𝜓) → (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))

Theorembj-alexim 33188 Closed form of aleximi 1875 (with a shorter proof, so aleximi 1875 could be deduced from it --- exim 1877 would have to be proved first, but its proof is shorter (currently almost a subproof of aleximi 1875)). (Contributed by BJ, 8-Nov-2021.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)))

Theorembj-nexdh 33189 Closed form of nexdh 1910 (actually, its general instance). (Contributed by BJ, 6-May-2019.)
(∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓)))

Theorembj-nexdh2 33190 Uncurried (imported) form of bj-nexdh 33189. (Contributed by BJ, 6-May-2019.)
((∀𝑥(𝜑 → ¬ 𝜓) ∧ (𝜒 → ∀𝑥𝜑)) → (𝜒 → ¬ ∃𝑥𝜓))

Theorembj-hbxfrbi 33191 Closed form of hbxfrbi 1868. Notes: it is less important than nfbiit 1895; it requires sp 2167 (unlike nfbiit 1895); there is an obvious version with (∃𝑥𝜑𝜑) instead. (Contributed by BJ, 6-May-2019.)
(∀𝑥(𝜑𝜓) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))

Theorembj-exlime 33192 Variant of exlimih 2263 where the nonfreeness of 𝑥 in 𝜓 is expressed using an existential quantifier, thus requiring fewer axioms. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝜓𝜓)    &   (𝜑𝜓)       (∃𝑥𝜑𝜓)

Theorembj-exnalimn 33193 A transformation of quantifiers and logical connectives. The general statement that equs3 2006 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 1953. I propose to move to the main part: bj-exnalimn 33193, bj-exalim 33194, bj-exalimi 33195, bj-exalims 33196, bj-exalimsi 33197, bj-ax12i 33199, bj-ax12wlem 33203, bj-ax12w 33258, and remove equs3 2006. A new label is needed for bj-ax12i 33199 and label suggestions are welcome for the others. I also propose to change ¬ ∀𝑥¬ to 𝑥 in speimfw 2007 and spimfw 2009 (other spim* theorems use 𝑥 and very few theorems in set.mm use ¬ ∀𝑥¬). (Contributed by BJ, 29-Sep-2019.)

(∃𝑥(𝜑𝜓) ↔ ¬ ∀𝑥(𝜑 → ¬ 𝜓))

Theorembj-exalim 33194 Distributing quantifiers over a double implication. (Contributed by BJ, 8-Nov-2021.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)))

Theorembj-exalimi 33195 An inference for distributing quantifiers over a double implication. The canonical derivation from its closed form bj-exalim 33194 (using mpg 1841) has fewer essential steps, but more steps in total (yielding a longer compressed proof). (Almost) the general statement that speimfw 2007 proves. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))       (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))

Theorembj-exalims 33196 Distributing quantifiers over a double implication. (Almost) the general statement that spimfw 2009 proves. (Contributed by BJ, 29-Sep-2019.)
(∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))       (∀𝑥(𝜑 → (𝜓𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓𝜒)))

Theorembj-exalimsi 33197 An inference for distributing quantifiers over a double implication. (Almost) the general statement that spimfw 2009 proves. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))    &   (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))       (∃𝑥𝜑 → (∀𝑥𝜓𝜒))

Theorembj-ax12ig 33198 A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 33199. (Contributed by BJ, 19-Dec-2020.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))

Theorembj-ax12i 33199 A weakening of bj-ax12ig 33198 that is sufficient to prove a weak form of the axiom of substitution ax-12 2163. The general statement of which ax12i 2010 is an instance. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))    &   (𝜒 → ∀𝑥𝜒)       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))

Theorembj-nfimt 33200 Closed form of nfim 1943 and curried (exported) form of nfimt 1941. (Contributed by BJ, 20-Oct-2021.)
(Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑𝜓)))

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