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Theorem List for Metamath Proof Explorer - 43501-43600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremaifftbifffaibif 43501 Given a is equivalent to T., Given b is equivalent to F., there exists a proof for that a implies b is false. (Contributed by Jarvin Udandy, 7-Sep-2020.)
(𝜑 ↔ ⊤)    &   (𝜓 ↔ ⊥)       ((𝜑𝜓) ↔ ⊥)

Theoremaifftbifffaibifff 43502 Given a is equivalent to T., Given b is equivalent to F., there exists a proof for that a iff b is false. (Contributed by Jarvin Udandy, 7-Sep-2020.)
(𝜑 ↔ ⊤)    &   (𝜓 ↔ ⊥)       ((𝜑𝜓) ↔ ⊥)

Theorematnaiana 43503 Given a, it is not the case a implies a self contradiction. (Contributed by Jarvin Udandy, 7-Sep-2020.)
𝜑        ¬ (𝜑 → (𝜑 ∧ ¬ 𝜑))

Theoremainaiaandna 43504 Given a, a implies it is not the case a implies a self contradiction. (Contributed by Jarvin Udandy, 7-Sep-2020.)
𝜑       (𝜑 → ¬ (𝜑 → (𝜑 ∧ ¬ 𝜑)))

Theoremabcdta 43505 Given (((a and b) and c) and d), there exists a proof for a. (Contributed by Jarvin Udandy, 3-Sep-2016.)
(((𝜑𝜓) ∧ 𝜒) ∧ 𝜃)       𝜑

Theoremabcdtb 43506 Given (((a and b) and c) and d), there exists a proof for b. (Contributed by Jarvin Udandy, 3-Sep-2016.)
(((𝜑𝜓) ∧ 𝜒) ∧ 𝜃)       𝜓

Theoremabcdtc 43507 Given (((a and b) and c) and d), there exists a proof for c. (Contributed by Jarvin Udandy, 3-Sep-2016.)
(((𝜑𝜓) ∧ 𝜒) ∧ 𝜃)       𝜒

Theoremabcdtd 43508 Given (((a and b) and c) and d), there exists a proof for d. (Contributed by Jarvin Udandy, 3-Sep-2016.)
(((𝜑𝜓) ∧ 𝜒) ∧ 𝜃)       𝜃

Theoremabciffcbatnabciffncba 43509 Operands in a biconditional expression converted negated. Additionally biconditional converted to show antecedent implies sequent. Closed form. (Contributed by Jarvin Udandy, 7-Sep-2020.)
(¬ ((𝜑𝜓) ∧ 𝜒) → ¬ ((𝜒𝜓) ∧ 𝜑))

Theoremabciffcbatnabciffncbai 43510 Operands in a biconditional expression converted negated. Additionally biconditional converted to show antecedent implies sequent. (Contributed by Jarvin Udandy, 7-Sep-2020.)
(((𝜑𝜓) ∧ 𝜒) ↔ ((𝜒𝜓) ∧ 𝜑))       (¬ ((𝜑𝜓) ∧ 𝜒) → ¬ ((𝜒𝜓) ∧ 𝜑))

Theoremnabctnabc 43511 not ( a -> ( b /\ c ) ) we can show: not a implies ( b /\ c ). (Contributed by Jarvin Udandy, 7-Sep-2020.)
¬ (𝜑 → (𝜓𝜒))       𝜑 → (𝜓𝜒))

Theoremjabtaib 43512 For when pm3.4 lacks a pm3.4i. (Contributed by Jarvin Udandy, 9-Sep-2020.)
(𝜑𝜓)       (𝜑𝜓)

Theoremonenotinotbothi 43513 From one negated implication it is not the case its nonnegated form and a random others are both true. (Contributed by Jarvin Udandy, 11-Sep-2020.)
¬ (𝜑𝜓)        ¬ ((𝜑𝜓) ∧ (𝜒𝜃))

Theoremtwonotinotbothi 43514 From these two negated implications it is not the case their nonnegated forms are both true. (Contributed by Jarvin Udandy, 11-Sep-2020.)
¬ (𝜑𝜓)    &    ¬ (𝜒𝜃)        ¬ ((𝜑𝜓) ∧ (𝜒𝜃))

Theoremclifte 43515 show d is the same as an if-else involving a,b. (Contributed by Jarvin Udandy, 20-Sep-2020.)
(𝜑 ∧ ¬ 𝜒)    &   𝜃       (𝜃 ↔ ((𝜑 ∧ ¬ 𝜒) ∨ (𝜓𝜒)))

Theoremcliftet 43516 show d is the same as an if-else involving a,b. (Contributed by Jarvin Udandy, 20-Sep-2020.)
(𝜑𝜒)    &   𝜃       (𝜃 ↔ ((𝜑𝜒) ∨ (𝜓 ∧ ¬ 𝜒)))

Theoremclifteta 43517 show d is the same as an if-else involving a,b. (Contributed by Jarvin Udandy, 20-Sep-2020.)
((𝜑 ∧ ¬ 𝜒) ∨ (𝜓𝜒))    &   𝜃       (𝜃 ↔ ((𝜑 ∧ ¬ 𝜒) ∨ (𝜓𝜒)))

Theoremcliftetb 43518 show d is the same as an if-else involving a,b. (Contributed by Jarvin Udandy, 20-Sep-2020.)
((𝜑𝜒) ∨ (𝜓 ∧ ¬ 𝜒))    &   𝜃       (𝜃 ↔ ((𝜑𝜒) ∨ (𝜓 ∧ ¬ 𝜒)))

Theoremconfun 43519 Given the hypotheses there exists a proof for (c implies ( d iff a ) ). (Contributed by Jarvin Udandy, 6-Sep-2020.)
𝜑    &   (𝜒𝜓)    &   (𝜒𝜃)    &   (𝜑 → (𝜑𝜓))       (𝜒 → (𝜃𝜑))

Theoremconfun2 43520 Confun simplified to two propositions. (Contributed by Jarvin Udandy, 6-Sep-2020.)
(𝜓𝜑)    &   (𝜓 → ¬ (𝜓 → (𝜓 ∧ ¬ 𝜓)))    &   ((𝜓𝜑) → ((𝜓𝜑) → 𝜑))       (𝜓 → (¬ (𝜓 → (𝜓 ∧ ¬ 𝜓)) ↔ (𝜓𝜑)))

Theoremconfun3 43521 Confun's more complex form where both a,d have been "defined". (Contributed by Jarvin Udandy, 6-Sep-2020.)
(𝜑 ↔ (𝜒𝜓))    &   (𝜃 ↔ ¬ (𝜒 → (𝜒 ∧ ¬ 𝜒)))    &   (𝜒𝜓)    &   (𝜒 → ¬ (𝜒 → (𝜒 ∧ ¬ 𝜒)))    &   ((𝜒𝜓) → ((𝜒𝜓) → 𝜓))       (𝜒 → (¬ (𝜒 → (𝜒 ∧ ¬ 𝜒)) ↔ (𝜒𝜓)))

Theoremconfun4 43522 An attempt at derivative. Resisted simplest path to a proof. (Contributed by Jarvin Udandy, 6-Sep-2020.)
𝜑    &   ((𝜑𝜓) → 𝜓)    &   (𝜓 → (𝜑𝜒))    &   ((𝜒𝜃) → ((𝜑𝜃) ↔ 𝜓))    &   (𝜏 ↔ (𝜒𝜃))    &   (𝜂 ↔ ¬ (𝜒 → (𝜒 ∧ ¬ 𝜒)))    &   𝜓    &   (𝜒𝜃)       (𝜒 → (𝜓𝜏))

Theoremconfun5 43523 An attempt at derivative. Resisted simplest path to a proof. Interesting that ch, th, ta, et were all provable. (Contributed by Jarvin Udandy, 7-Sep-2020.)
𝜑    &   ((𝜑𝜓) → 𝜓)    &   (𝜓 → (𝜑𝜒))    &   ((𝜒𝜃) → ((𝜑𝜃) ↔ 𝜓))    &   (𝜏 ↔ (𝜒𝜃))    &   (𝜂 ↔ ¬ (𝜒 → (𝜒 ∧ ¬ 𝜒)))    &   𝜓    &   (𝜒𝜃)       (𝜒 → (𝜂𝜏))

Theoremplcofph 43524 Given, a,b and a "definition" for c, c is demonstrated. (Contributed by Jarvin Udandy, 8-Sep-2020.)
(𝜒 ↔ ((((𝜑𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))))    &   𝜑    &   𝜓       𝜒

Theorempldofph 43525 Given, a,b c, d, "definition" for e, e is demonstrated. (Contributed by Jarvin Udandy, 8-Sep-2020.)
(𝜏 ↔ ((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃))))    &   𝜑    &   𝜓    &   𝜒    &   𝜃       𝜏

Theoremplvcofph 43526 Given, a,b,d, and "definitions" for c, e, f: f is demonstrated. (Contributed by Jarvin Udandy, 8-Sep-2020.)
(𝜒 ↔ ((((𝜑𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))))    &   (𝜏 ↔ ((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃))))    &   (𝜂 ↔ (𝜒𝜏))    &   𝜑    &   𝜓    &   𝜃       𝜂

Theoremplvcofphax 43527 Given, a,b,d, and "definitions" for c, e, f, g: g is demonstrated. (Contributed by Jarvin Udandy, 8-Sep-2020.)
(𝜒 ↔ ((((𝜑𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))))    &   (𝜏 ↔ ((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃))))    &   (𝜂 ↔ (𝜒𝜏))    &   𝜑    &   𝜓    &   𝜃    &   (𝜁 ↔ ¬ (𝜓 ∧ ¬ 𝜏))       𝜁

Theoremplvofpos 43528 rh is derivable because ONLY one of ch, th, ta, et is implied by mu. (Contributed by Jarvin Udandy, 11-Sep-2020.)
(𝜒 ↔ (¬ 𝜑 ∧ ¬ 𝜓))    &   (𝜃 ↔ (¬ 𝜑𝜓))    &   (𝜏 ↔ (𝜑 ∧ ¬ 𝜓))    &   (𝜂 ↔ (𝜑𝜓))    &   (𝜁 ↔ (((((¬ ((𝜇𝜒) ∧ (𝜇𝜃)) ∧ ¬ ((𝜇𝜒) ∧ (𝜇𝜏))) ∧ ¬ ((𝜇𝜒) ∧ (𝜒𝜂))) ∧ ¬ ((𝜇𝜃) ∧ (𝜇𝜏))) ∧ ¬ ((𝜇𝜃) ∧ (𝜇𝜂))) ∧ ¬ ((𝜇𝜏) ∧ (𝜇𝜂))))    &   (𝜎 ↔ (((𝜇𝜒) ∨ (𝜇𝜃)) ∨ ((𝜇𝜏) ∨ (𝜇𝜂))))    &   (𝜌 ↔ (𝜁𝜎))    &   𝜁    &   𝜎       𝜌

Theoremmdandyv0 43529 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊤)    &   (𝜒 ↔ ⊥)    &   (𝜃 ↔ ⊥)    &   (𝜏 ↔ ⊥)    &   (𝜂 ↔ ⊥)       ((((𝜒𝜑) ∧ (𝜃𝜑)) ∧ (𝜏𝜑)) ∧ (𝜂𝜑))

Theoremmdandyv1 43530 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊤)    &   (𝜒 ↔ ⊤)    &   (𝜃 ↔ ⊥)    &   (𝜏 ↔ ⊥)    &   (𝜂 ↔ ⊥)       ((((𝜒𝜓) ∧ (𝜃𝜑)) ∧ (𝜏𝜑)) ∧ (𝜂𝜑))

Theoremmdandyv2 43531 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊤)    &   (𝜒 ↔ ⊥)    &   (𝜃 ↔ ⊤)    &   (𝜏 ↔ ⊥)    &   (𝜂 ↔ ⊥)       ((((𝜒𝜑) ∧ (𝜃𝜓)) ∧ (𝜏𝜑)) ∧ (𝜂𝜑))

Theoremmdandyv3 43532 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊤)    &   (𝜒 ↔ ⊤)    &   (𝜃 ↔ ⊤)    &   (𝜏 ↔ ⊥)    &   (𝜂 ↔ ⊥)       ((((𝜒𝜓) ∧ (𝜃𝜓)) ∧ (𝜏𝜑)) ∧ (𝜂𝜑))

Theoremmdandyv4 43533 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊤)    &   (𝜒 ↔ ⊥)    &   (𝜃 ↔ ⊥)    &   (𝜏 ↔ ⊤)    &   (𝜂 ↔ ⊥)       ((((𝜒𝜑) ∧ (𝜃𝜑)) ∧ (𝜏𝜓)) ∧ (𝜂𝜑))

Theoremmdandyv5 43534 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊤)    &   (𝜒 ↔ ⊤)    &   (𝜃 ↔ ⊥)    &   (𝜏 ↔ ⊤)    &   (𝜂 ↔ ⊥)       ((((𝜒𝜓) ∧ (𝜃𝜑)) ∧ (𝜏𝜓)) ∧ (𝜂𝜑))

Theoremmdandyv6 43535 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊤)    &   (𝜒 ↔ ⊥)    &   (𝜃 ↔ ⊤)    &   (𝜏 ↔ ⊤)    &   (𝜂 ↔ ⊥)       ((((𝜒𝜑) ∧ (𝜃𝜓)) ∧ (𝜏𝜓)) ∧ (𝜂𝜑))

Theoremmdandyv7 43536 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊤)    &   (𝜒 ↔ ⊤)    &   (𝜃 ↔ ⊤)    &   (𝜏 ↔ ⊤)    &   (𝜂 ↔ ⊥)       ((((𝜒𝜓) ∧ (𝜃𝜓)) ∧ (𝜏𝜓)) ∧ (𝜂𝜑))

Theoremmdandyv8 43537 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊤)    &   (𝜒 ↔ ⊥)    &   (𝜃 ↔ ⊥)    &   (𝜏 ↔ ⊥)    &   (𝜂 ↔ ⊤)       ((((𝜒𝜑) ∧ (𝜃𝜑)) ∧ (𝜏𝜑)) ∧ (𝜂𝜓))

Theoremmdandyv9 43538 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊤)    &   (𝜒 ↔ ⊤)    &   (𝜃 ↔ ⊥)    &   (𝜏 ↔ ⊥)    &   (𝜂 ↔ ⊤)       ((((𝜒𝜓) ∧ (𝜃𝜑)) ∧ (𝜏𝜑)) ∧ (𝜂𝜓))

Theoremmdandyv10 43539 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊤)    &   (𝜒 ↔ ⊥)    &   (𝜃 ↔ ⊤)    &   (𝜏 ↔ ⊥)    &   (𝜂 ↔ ⊤)       ((((𝜒𝜑) ∧ (𝜃𝜓)) ∧ (𝜏𝜑)) ∧ (𝜂𝜓))

Theoremmdandyv11 43540 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊤)    &   (𝜒 ↔ ⊤)    &   (𝜃 ↔ ⊤)    &   (𝜏 ↔ ⊥)    &   (𝜂 ↔ ⊤)       ((((𝜒𝜓) ∧ (𝜃𝜓)) ∧ (𝜏𝜑)) ∧ (𝜂𝜓))

Theoremmdandyv12 43541 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊤)    &   (𝜒 ↔ ⊥)    &   (𝜃 ↔ ⊥)    &   (𝜏 ↔ ⊤)    &   (𝜂 ↔ ⊤)       ((((𝜒𝜑) ∧ (𝜃𝜑)) ∧ (𝜏𝜓)) ∧ (𝜂𝜓))

Theoremmdandyv13 43542 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊤)    &   (𝜒 ↔ ⊤)    &   (𝜃 ↔ ⊥)    &   (𝜏 ↔ ⊤)    &   (𝜂 ↔ ⊤)       ((((𝜒𝜓) ∧ (𝜃𝜑)) ∧ (𝜏𝜓)) ∧ (𝜂𝜓))

Theoremmdandyv14 43543 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊤)    &   (𝜒 ↔ ⊥)    &   (𝜃 ↔ ⊤)    &   (𝜏 ↔ ⊤)    &   (𝜂 ↔ ⊤)       ((((𝜒𝜑) ∧ (𝜃𝜓)) ∧ (𝜏𝜓)) ∧ (𝜂𝜓))

Theoremmdandyv15 43544 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016.)
(𝜑 ↔ ⊥)    &   (𝜓 ↔ ⊤)    &   (𝜒 ↔ ⊤)    &   (𝜃 ↔ ⊤)    &   (𝜏 ↔ ⊤)    &   (𝜂 ↔ ⊤)       ((((𝜒𝜓) ∧ (𝜃𝜓)) ∧ (𝜏𝜓)) ∧ (𝜂𝜓))

Theoremmdandyvr0 43545 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜑)    &   (𝜏𝜑)    &   (𝜂𝜑)       ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜁)) ∧ (𝜂𝜁))

Theoremmdandyvr1 43546 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜑)    &   (𝜏𝜑)    &   (𝜂𝜑)       ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜁)) ∧ (𝜂𝜁))

Theoremmdandyvr2 43547 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜑)    &   (𝜂𝜑)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜁))

Theoremmdandyvr3 43548 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜑)    &   (𝜂𝜑)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜁))

Theoremmdandyvr4 43549 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvr5 43550 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvr6 43551 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvr7 43552 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvr8 43553 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜑)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvr9 43554 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜑)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvr10 43555 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvr11 43556 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvr12 43557 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

Theoremmdandyvr13 43558 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

Theoremmdandyvr14 43559 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

Theoremmdandyvr15 43560 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

Theoremmdandyvrx0 43561 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜑)    &   (𝜏𝜑)    &   (𝜂𝜑)       ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜁)) ∧ (𝜂𝜁))

Theoremmdandyvrx1 43562 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜑)    &   (𝜏𝜑)    &   (𝜂𝜑)       ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜁)) ∧ (𝜂𝜁))

Theoremmdandyvrx2 43563 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜑)    &   (𝜂𝜑)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜁))

Theoremmdandyvrx3 43564 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜑)    &   (𝜂𝜑)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜁))

Theoremmdandyvrx4 43565 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvrx5 43566 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvrx6 43567 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvrx7 43568 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvrx8 43569 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜑)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvrx9 43570 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜑)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvrx10 43571 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvrx11 43572 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvrx12 43573 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

Theoremmdandyvrx13 43574 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

Theoremmdandyvrx14 43575 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

Theoremmdandyvrx15 43576 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

TheoremH15NH16TH15IH16 43577 Given 15 hypotheses and a 16th hypothesis, there exists a proof the 15 imply the 16th. (Contributed by Jarvin Udandy, 8-Sep-2016.)
𝜑    &   𝜓    &   𝜒    &   𝜃    &   𝜏    &   𝜂    &   𝜁    &   𝜎    &   𝜌    &   𝜇    &   𝜆    &   𝜅    &   jph    &   jps    &   jch    &   jth       (((((((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) ∧ jph) ∧ jps) ∧ jch) → jth)

Theoremdandysum2p2e4 43578

CONTRADICTION PROVED AT 1 + 1 = 2 .

Given the right hypotheses we can prove a dandysum of 2+2=4. The qed step is the value '4' in Decimal BEING IMPLIED by the hypotheses.

Note: Values that when added which exceed a 4bit value are not supported.

Note: Digits begin from left (least) to right (greatest). e.g. 1000 would be '1', 0100 would be '2'. 0010 would be '4'.

How to perceive the hypotheses' bits in order: ( th <-> F. ), ( ta <-> F. ) Would be input value X's first bit, and input value Y's first bit.

( et <-> F ), ( ze <-> F. ) would be input value X's second bit, and input value Y's second bit. (Contributed by Jarvin Udandy, 6-Sep-2016.)

(𝜑 ↔ (𝜃𝜏))    &   (𝜓 ↔ (𝜂𝜁))    &   (𝜒 ↔ (𝜎𝜌))    &   (𝜃 ↔ ⊥)    &   (𝜏 ↔ ⊥)    &   (𝜂 ↔ ⊤)    &   (𝜁 ↔ ⊤)    &   (𝜎 ↔ ⊥)    &   (𝜌 ↔ ⊥)    &   (𝜇 ↔ ⊥)    &   (𝜆 ↔ ⊥)    &   (𝜅 ↔ ((𝜃𝜏) ⊻ (𝜃𝜏)))    &   (jph ↔ ((𝜂𝜁) ∨ 𝜑))    &   (jps ↔ ((𝜎𝜌) ∨ 𝜓))    &   (jch ↔ ((𝜇𝜆) ∨ 𝜒))       ((((((((((((((((𝜑 ↔ (𝜃𝜏)) ∧ (𝜓 ↔ (𝜂𝜁))) ∧ (𝜒 ↔ (𝜎𝜌))) ∧ (𝜃 ↔ ⊥)) ∧ (𝜏 ↔ ⊥)) ∧ (𝜂 ↔ ⊤)) ∧ (𝜁 ↔ ⊤)) ∧ (𝜎 ↔ ⊥)) ∧ (𝜌 ↔ ⊥)) ∧ (𝜇 ↔ ⊥)) ∧ (𝜆 ↔ ⊥)) ∧ (𝜅 ↔ ((𝜃𝜏) ⊻ (𝜃𝜏)))) ∧ (jph ↔ ((𝜂𝜁) ∨ 𝜑))) ∧ (jps ↔ ((𝜎𝜌) ∨ 𝜓))) ∧ (jch ↔ ((𝜇𝜆) ∨ 𝜒))) → ((((𝜅 ↔ ⊥) ∧ (jph ↔ ⊥)) ∧ (jps ↔ ⊤)) ∧ (jch ↔ ⊥)))

Theoremmdandysum2p2e4 43579 CONTRADICTION PROVED AT 1 + 1 = 2 . Luckily Mario Carneiro did a successful version of his own.

See Mario's Relevant Work: 1.3.14 Half adder and full adder in propositional calculus.

Given the right hypotheses we can prove a dandysum of 2+2=4. The qed step is the value '4' in Decimal BEING IMPLIED by the hypotheses.

Note: Values that when added which exceed a 4bit value are not supported.

Note: Digits begin from left (least) to right (greatest). e.g. 1000 would be '1', 0100 would be '2'. 0010 would be '4'.

How to perceive the hypotheses' bits in order: ( th <-> F. ), ( ta <-> F. ) Would be input value X's first bit, and input value Y's first bit.

( et <-> F. ), ( ze <-> F. ) would be input value X's second bit, and input value Y's second bit.

In mdandysum2p2e4, one might imagine what jth or jta could be then do the math with their truths. Also limited to the restriction jth, jta are having opposite truths equivalent to the stated truth constants.

(Contributed by Jarvin Udandy, 6-Sep-2016.)

(jth ↔ ⊥)    &   (jta ↔ ⊤)    &   (𝜑 ↔ (𝜃𝜏))    &   (𝜓 ↔ (𝜂𝜁))    &   (𝜒 ↔ (𝜎𝜌))    &   (𝜃jth)    &   (𝜏jth)    &   (𝜂jta)    &   (𝜁jta)    &   (𝜎jth)    &   (𝜌jth)    &   (𝜇jth)    &   (𝜆jth)    &   (𝜅 ↔ ((𝜃𝜏) ⊻ (𝜃𝜏)))    &   (jph ↔ ((𝜂𝜁) ∨ 𝜑))    &   (jps ↔ ((𝜎𝜌) ∨ 𝜓))    &   (jch ↔ ((𝜇𝜆) ∨ 𝜒))       ((((((((((((((((𝜑 ↔ (𝜃𝜏)) ∧ (𝜓 ↔ (𝜂𝜁))) ∧ (𝜒 ↔ (𝜎𝜌))) ∧ (𝜃 ↔ ⊥)) ∧ (𝜏 ↔ ⊥)) ∧ (𝜂 ↔ ⊤)) ∧ (𝜁 ↔ ⊤)) ∧ (𝜎 ↔ ⊥)) ∧ (𝜌 ↔ ⊥)) ∧ (𝜇 ↔ ⊥)) ∧ (𝜆 ↔ ⊥)) ∧ (𝜅 ↔ ((𝜃𝜏) ⊻ (𝜃𝜏)))) ∧ (jph ↔ ((𝜂𝜁) ∨ 𝜑))) ∧ (jps ↔ ((𝜎𝜌) ∨ 𝜓))) ∧ (jch ↔ ((𝜇𝜆) ∨ 𝜒))) → ((((𝜅 ↔ ⊥) ∧ (jph ↔ ⊥)) ∧ (jps ↔ ⊤)) ∧ (jch ↔ ⊥)))

Theoremadh-jarrsc 43580 Replacement of a nested antecedent with an outer antecedent. Commuted simplificated form of elimination of a nested antecedent. Also holds intuitionistically. Polish prefix notation: CCCpqrCsCqr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜃 → (𝜓𝜒)))

20.40.1  Minimal implicational calculus

Minimal implicational calculus, or intuitionistic implicational calculus, or positive implicational calculus, is the implicational fragment of minimal calculus (which is also the implicational fragment of intuitionistic calculus and of positive calculus). It is sometimes called "C-pure intuitionism" since the letter C is used to denote implication in Polish prefix notation. It can be axiomatized by the inference rule of modus ponens ax-mp 5 together with the axioms { ax-1 6, ax-2 7 } (sometimes written KS), or with { imim1 83, ax-1 6, pm2.43 56 } (written B'KW), or with { imim2 58, pm2.04 90, ax-1 6, pm2.43 56 } (written BCKW), or with the single axiom adh-minim 43581, or with the single axiom adh-minimp 43593. This section proves first adh-minim 43581 from { ax-1 6, ax-2 7 }, followed by the converse, due to Ivo Thomas; and then it proves adh-minimp 43593 from { ax-1 6, ax-2 7 }, also followed by the converse, also due to Ivo Thomas.

Sources for this section are * Carew Arthur Meredith, A single axiom of positive logic, The Journal of Computing Systems, volume 1, issue 3, July 1953, pages 169--170; * Ivo Thomas, On Meredith's sole positive axiom, Notre Dame Journal of Formal Logic, volume XV, number 3, July 1974, page 477, in which the derivations of { ax-1 6, ax-2 7 } from adh-minim 43581 are shortened (compared to Meredith's derivations in the aforementioned paper); * Carew Arthur Meredith and Arthur Norman Prior, Notes on the axiomatics of the propositional calculus, Notre Dame Journal of Formal Logic, volume IV, number 3, July 1963, pages 171--187; and * the webpage https://web.ics.purdue.edu/~dulrich/C-pure-intuitionism-page.htm 43581 on Dolph Edward "Ted" Ulrich's website, where these and other single axioms for the minimal implicational calculus are listed.

This entire section also holds intuitionistically.

Users of the Polish prefix notation also often use a compact notation for proof derivations known as the D-notation where "D" stands for "condensed Detachment". For instance, "D21" means detaching ax-1 6 from ax-2 7, that is, using modus ponens ax-mp 5 with ax-1 6 as minor premise and ax-2 7 as major premise. When the numbered lemmas surpass 10, dots are added between the numbers. D-strings are accepted by the grammar Dundotted := digit | "D" Dundotted Dundotted ; Ddotted := digit + | "D" Ddotted "." Ddotted ; Dstr := Dundotted | Ddotted .

(Contributed by BJ, 11-Apr-2021.) (Revised by ADH, 10-Nov-2023.)

Theoremadh-minim 43581 A single axiom for minimal implicational calculus, due to Meredith. Other single axioms of the same length are known, but it is thought to be the minimal length. This is the axiom from Carew Arthur Meredith, A single axiom of positive logic, The Journal of Computing Systems, volume 1, issue 3, July 1953, pages 169--170. A two-line review by Alonzo Church of this article can be found in The Journal of Symbolic Logic, volume 19, issue 2, June 1954, page 144, https://doi.org/10.2307/2268914. Known as "HI-1" on Dolph Edward "Ted" Ulrich's web page. In the next 6 lemmas and 3 theorems, ax-1 6 and ax-2 7 are derived from this single axiom in 16 detachments (instances of ax-mp 5) in total. Polish prefix notation: CCCpqrCsCCqCrtCqt . (Contributed by ADH, 10-Nov-2023.)
(((𝜑𝜓) → 𝜒) → (𝜃 → ((𝜓 → (𝜒𝜏)) → (𝜓𝜏))))

Theoremadh-minim-ax1-ax2-lem1 43582 First lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 43581 and ax-mp 5. Polish prefix notation: CpCCqCCrCCsCqtCstuCqu . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ((𝜓 → ((𝜒 → ((𝜃 → (𝜓𝜏)) → (𝜃𝜏))) → 𝜂)) → (𝜓𝜂)))

Theoremadh-minim-ax1-ax2-lem2 43583 Second lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 43581 and ax-mp 5. Polish prefix notation: CCpCCqCCrCpsCrstCpt . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → ((𝜓 → ((𝜒 → (𝜑𝜃)) → (𝜒𝜃))) → 𝜏)) → (𝜑𝜏))

Theoremadh-minim-ax1-ax2-lem3 43584 Third lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 43581 and ax-mp 5. Polish prefix notation: CCpCqrCqCsCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜃 → (𝜑𝜒))))

Theoremadh-minim-ax1-ax2-lem4 43585 Fourth lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 43581 and ax-mp 5. Polish prefix notation: CCCpqrCCqCrsCqs . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → ((𝜓 → (𝜒𝜃)) → (𝜓𝜃)))

Theoremadh-minim-ax1 43586 Derivation of ax-1 6 from adh-minim 43581 and ax-mp 5. Carew Arthur Meredith derived ax-1 6 in A single axiom of positive logic, The Journal of Computing Systems, volume 1, issue 3, July 1953, pages 169--170. However, here we follow the shortened derivation by Ivo Thomas, On Meredith's sole positive axiom, Notre Dame Journal of Formal Logic, volume XV, number 3, July 1974, page 477. Polish prefix notation: CpCqp . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))

Theoremadh-minim-ax2-lem5 43587 Fifth lemma for the derivation of ax-2 7 from adh-minim 43581 and ax-mp 5. Polish prefix notation: CpCCCqrsCCrCstCrt . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (((𝜓𝜒) → 𝜃) → ((𝜒 → (𝜃𝜏)) → (𝜒𝜏))))

Theoremadh-minim-ax2-lem6 43588 Sixth lemma for the derivation of ax-2 7 from adh-minim 43581 and ax-mp 5. Polish prefix notation: CCpCCCCqrsCCrCstCrtuCpu . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → ((((𝜓𝜒) → 𝜃) → ((𝜒 → (𝜃𝜏)) → (𝜒𝜏))) → 𝜂)) → (𝜑𝜂))

Theoremadh-minim-ax2c 43589 Derivation of a commuted form of ax-2 7 from adh-minim 43581 and ax-mp 5. Polish prefix notation: CCpqCCpCqrCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))

Theoremadh-minim-ax2 43590 Derivation of ax-2 7 from adh-minim 43581 and ax-mp 5. Carew Arthur Meredith derived ax-2 7 in A single axiom of positive logic, The Journal of Computing Systems, volume 1, issue 3, July 1953, pages 169--170. However, here we follow the shortened derivation by Ivo Thomas, On Meredith's sole positive axiom, Notre Dame Journal of Formal Logic, volume XV, number 3, July 1974, page 477. Polish prefix notation: CCpCqrCCpqCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))

Theoremadh-minim-idALT 43591 Derivation of id 22 (reflexivity of implication, PM *2.08 WhiteheadRussell p. 101) from adh-minim-ax1 43586, adh-minim-ax2 43590, and ax-mp 5. It uses the derivation written DD211 in D-notation. (See head comment for an explanation.) Polish prefix notation: Cpp . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜑)

Theoremadh-minim-pm2.43 43592 Derivation of pm2.43 56 WhiteheadRussell p. 106 (also called "hilbert" or "W") from adh-minim-ax1 43586, adh-minim-ax2 43590, and ax-mp 5. It uses the derivation written DD22D21 in D-notation. (See head comment for an explanation.) (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜑𝜓)) → (𝜑𝜓))

Theoremadh-minimp 43593 Another single axiom for minimal implicational calculus, due to Meredith. Other single axioms of the same length are known, but it is thought to be the minimal length. Among single axioms of this length, it is the one with simplest antecedents (i.e., in the corresponding ordering of binary trees which first compares left subtrees, it is the first one). Known as "HI-2" on Dolph Edward "Ted" Ulrich's web page. In the next 4 lemmas and 5 theorems, ax-1 6 and ax-2 7 are derived from this other single axiom in 20 detachments (instances of ax-mp 5) in total. Polish prefix notation: CpCCqrCCCsqCrtCqt ; or CtCCpqCCCspCqrCpr in Carew Arthur Meredith and Arthur Norman Prior, Notes on the axiomatics of the propositional calculus, Notre Dame Journal of Formal Logic, volume IV, number 3, July 1963, pages 171--187, on page 180. (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.)
(𝜑 → ((𝜓𝜒) → (((𝜃𝜓) → (𝜒𝜏)) → (𝜓𝜏))))

Theoremadh-minimp-jarr-imim1-ax2c-lem1 43594 First lemma for the derivation of jarr 106, imim1 83, and a commuted form of ax-2 7, and indirectly ax-1 6 and ax-2 7, from adh-minimp 43593 and ax-mp 5. Polish prefix notation: CCpqCCCrpCqsCps . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (((𝜒𝜑) → (𝜓𝜃)) → (𝜑𝜃)))

Theoremadh-minimp-jarr-lem2 43595 Second lemma for the derivation of jarr 106, and indirectly ax-1 6, a commuted form of ax-2 7, and ax-2 7 proper, from adh-minimp 43593 and ax-mp 5. Polish prefix notation: CCCpqCCCrsCCCtrCsuCruvCqv . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → (((𝜒𝜃) → (((𝜏𝜒) → (𝜃𝜂)) → (𝜒𝜂))) → 𝜁)) → (𝜓𝜁))

Theoremadh-minimp-jarr-ax2c-lem3 43596 Third lemma for the derivation of jarr 106 and a commuted form of ax-2 7, and indirectly ax-1 6 and ax-2 7 proper , from adh-minimp 43593 and ax-mp 5. Polish prefix notation: CCCCpqCCCrpCqsCpstt . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑𝜓) → (((𝜒𝜑) → (𝜓𝜃)) → (𝜑𝜃))) → 𝜏) → 𝜏)

Theoremadh-minimp-sylsimp 43597 Derivation of jarr 106 (also called "syll-simp") from minimp 1623 and ax-mp 5. Polish prefix notation: CCCpqrCqr . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜓𝜒))

Theoremadh-minimp-ax1 43598 Derivation of ax-1 6 from adh-minimp 43593 and ax-mp 5. Polish prefix notation: CpCqp . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))

Theoremadh-minimp-imim1 43599 Derivation of imim1 83 ("left antimonotonicity of implication", theorem *2.06 of [WhiteheadRussell] p. 100) from adh-minimp 43593 and ax-mp 5. Polish prefix notation: CCpqCCqrCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Theoremadh-minimp-ax2c 43600 Derivation of a commuted form of ax-2 7 from adh-minimp 43593 and ax-mp 5. Polish prefix notation: CCpqCCpCqrCpr . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))

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