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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | abnotataxb 43201 | Assuming not a, b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.) |
⊢ ¬ 𝜑 & ⊢ 𝜓 ⇒ ⊢ (𝜑 ⊻ 𝜓) | ||
Theorem | conimpf 43202 | Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 28-Aug-2016.) |
⊢ 𝜑 & ⊢ ¬ 𝜓 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 ↔ ⊥) | ||
Theorem | conimpfalt 43203 | Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 29-Aug-2016.) |
⊢ 𝜑 & ⊢ ¬ 𝜓 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 ↔ ⊥) | ||
Theorem | aistbisfiaxb 43204 | Given a is equivalent to T., Given b is equivalent to F. there exists a proof for a-xor-b. (Contributed by Jarvin Udandy, 31-Aug-2016.) |
⊢ (𝜑 ↔ ⊤) & ⊢ (𝜓 ↔ ⊥) ⇒ ⊢ (𝜑 ⊻ 𝜓) | ||
Theorem | aisfbistiaxb 43205 | Given a is equivalent to F., Given b is equivalent to T., there exists a proof for a-xor-b. (Contributed by Jarvin Udandy, 31-Aug-2016.) |
⊢ (𝜑 ↔ ⊥) & ⊢ (𝜓 ↔ ⊤) ⇒ ⊢ (𝜑 ⊻ 𝜓) | ||
Theorem | aifftbifffaibif 43206 | 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 ↔ ⊥) ⇒ ⊢ ((𝜑 → 𝜓) ↔ ⊥) | ||
Theorem | aifftbifffaibifff 43207 | 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 ↔ ⊥) ⇒ ⊢ ((𝜑 ↔ 𝜓) ↔ ⊥) | ||
Theorem | atnaiana 43208 | Given a, it is not the case a implies a self contradiction. (Contributed by Jarvin Udandy, 7-Sep-2020.) |
⊢ 𝜑 ⇒ ⊢ ¬ (𝜑 → (𝜑 ∧ ¬ 𝜑)) | ||
Theorem | ainaiaandna 43209 | 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))) | ||
Theorem | abcdta 43210 | Given (((a and b) and c) and d), there exists a proof for a. (Contributed by Jarvin Udandy, 3-Sep-2016.) |
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ⇒ ⊢ 𝜑 | ||
Theorem | abcdtb 43211 | Given (((a and b) and c) and d), there exists a proof for b. (Contributed by Jarvin Udandy, 3-Sep-2016.) |
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ⇒ ⊢ 𝜓 | ||
Theorem | abcdtc 43212 | Given (((a and b) and c) and d), there exists a proof for c. (Contributed by Jarvin Udandy, 3-Sep-2016.) |
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ⇒ ⊢ 𝜒 | ||
Theorem | abcdtd 43213 | Given (((a and b) and c) and d), there exists a proof for d. (Contributed by Jarvin Udandy, 3-Sep-2016.) |
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ⇒ ⊢ 𝜃 | ||
Theorem | abciffcbatnabciffncba 43214 | Operands in a biconditional expression converted negated. Additionally biconditional converted to show antecedent implies sequent. Closed form. (Contributed by Jarvin Udandy, 7-Sep-2020.) |
⊢ (¬ ((𝜑 ∧ 𝜓) ∧ 𝜒) → ¬ ((𝜒 ∧ 𝜓) ∧ 𝜑)) | ||
Theorem | abciffcbatnabciffncbai 43215 | Operands in a biconditional expression converted negated. Additionally biconditional converted to show antecedent implies sequent. (Contributed by Jarvin Udandy, 7-Sep-2020.) |
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜒 ∧ 𝜓) ∧ 𝜑)) ⇒ ⊢ (¬ ((𝜑 ∧ 𝜓) ∧ 𝜒) → ¬ ((𝜒 ∧ 𝜓) ∧ 𝜑)) | ||
Theorem | nabctnabc 43216 | not ( a -> ( b /\ c ) ) we can show: not a implies ( b /\ c ). (Contributed by Jarvin Udandy, 7-Sep-2020.) |
⊢ ¬ (𝜑 → (𝜓 ∧ 𝜒)) ⇒ ⊢ (¬ 𝜑 → (𝜓 ∧ 𝜒)) | ||
Theorem | jabtaib 43217 | For when pm3.4 lacks a pm3.4i. (Contributed by Jarvin Udandy, 9-Sep-2020.) |
⊢ (𝜑 ∧ 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | onenotinotbothi 43218 | 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.) |
⊢ ¬ (𝜑 → 𝜓) ⇒ ⊢ ¬ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) | ||
Theorem | twonotinotbothi 43219 | From these two negated implications it is not the case their nonnegated forms are both true. (Contributed by Jarvin Udandy, 11-Sep-2020.) |
⊢ ¬ (𝜑 → 𝜓) & ⊢ ¬ (𝜒 → 𝜃) ⇒ ⊢ ¬ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) | ||
Theorem | clifte 43220 | 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) ∨ (𝜓 ∧ 𝜒))) | ||
Theorem | cliftet 43221 | 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) ∨ (𝜓 ∧ ¬ 𝜒))) | ||
Theorem | clifteta 43222 | show d is the same as an if-else involving a,b. (Contributed by Jarvin Udandy, 20-Sep-2020.) |
⊢ ((𝜑 ∧ ¬ 𝜒) ∨ (𝜓 ∧ 𝜒)) & ⊢ 𝜃 ⇒ ⊢ (𝜃 ↔ ((𝜑 ∧ ¬ 𝜒) ∨ (𝜓 ∧ 𝜒))) | ||
Theorem | cliftetb 43223 | show d is the same as an if-else involving a,b. (Contributed by Jarvin Udandy, 20-Sep-2020.) |
⊢ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒)) & ⊢ 𝜃 ⇒ ⊢ (𝜃 ↔ ((𝜑 ∧ 𝜒) ∨ (𝜓 ∧ ¬ 𝜒))) | ||
Theorem | confun 43224 | Given the hypotheses there exists a proof for (c implies ( d iff a ) ). (Contributed by Jarvin Udandy, 6-Sep-2020.) |
⊢ 𝜑 & ⊢ (𝜒 → 𝜓) & ⊢ (𝜒 → 𝜃) & ⊢ (𝜑 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜒 → (𝜃 ↔ 𝜑)) | ||
Theorem | confun2 43225 | Confun simplified to two propositions. (Contributed by Jarvin Udandy, 6-Sep-2020.) |
⊢ (𝜓 → 𝜑) & ⊢ (𝜓 → ¬ (𝜓 → (𝜓 ∧ ¬ 𝜓))) & ⊢ ((𝜓 → 𝜑) → ((𝜓 → 𝜑) → 𝜑)) ⇒ ⊢ (𝜓 → (¬ (𝜓 → (𝜓 ∧ ¬ 𝜓)) ↔ (𝜓 → 𝜑))) | ||
Theorem | confun3 43226 | Confun's more complex form where both a,d have been "defined". (Contributed by Jarvin Udandy, 6-Sep-2020.) |
⊢ (𝜑 ↔ (𝜒 → 𝜓)) & ⊢ (𝜃 ↔ ¬ (𝜒 → (𝜒 ∧ ¬ 𝜒))) & ⊢ (𝜒 → 𝜓) & ⊢ (𝜒 → ¬ (𝜒 → (𝜒 ∧ ¬ 𝜒))) & ⊢ ((𝜒 → 𝜓) → ((𝜒 → 𝜓) → 𝜓)) ⇒ ⊢ (𝜒 → (¬ (𝜒 → (𝜒 ∧ ¬ 𝜒)) ↔ (𝜒 → 𝜓))) | ||
Theorem | confun4 43227 | An attempt at derivative. Resisted simplest path to a proof. (Contributed by Jarvin Udandy, 6-Sep-2020.) |
⊢ 𝜑 & ⊢ ((𝜑 → 𝜓) → 𝜓) & ⊢ (𝜓 → (𝜑 → 𝜒)) & ⊢ ((𝜒 → 𝜃) → ((𝜑 → 𝜃) ↔ 𝜓)) & ⊢ (𝜏 ↔ (𝜒 → 𝜃)) & ⊢ (𝜂 ↔ ¬ (𝜒 → (𝜒 ∧ ¬ 𝜒))) & ⊢ 𝜓 & ⊢ (𝜒 → 𝜃) ⇒ ⊢ (𝜒 → (𝜓 → 𝜏)) | ||
Theorem | confun5 43228 | 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 & ⊢ ((𝜑 → 𝜓) → 𝜓) & ⊢ (𝜓 → (𝜑 → 𝜒)) & ⊢ ((𝜒 → 𝜃) → ((𝜑 → 𝜃) ↔ 𝜓)) & ⊢ (𝜏 ↔ (𝜒 → 𝜃)) & ⊢ (𝜂 ↔ ¬ (𝜒 → (𝜒 ∧ ¬ 𝜒))) & ⊢ 𝜓 & ⊢ (𝜒 → 𝜃) ⇒ ⊢ (𝜒 → (𝜂 ↔ 𝜏)) | ||
Theorem | plcofph 43229 | Given, a,b and a "definition" for c, c is demonstrated. (Contributed by Jarvin Udandy, 8-Sep-2020.) |
⊢ (𝜒 ↔ ((((𝜑 ∧ 𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑)))) & ⊢ 𝜑 & ⊢ 𝜓 ⇒ ⊢ 𝜒 | ||
Theorem | pldofph 43230 | Given, a,b c, d, "definition" for e, e is demonstrated. (Contributed by Jarvin Udandy, 8-Sep-2020.) |
⊢ (𝜏 ↔ ((𝜒 → 𝜃) ∧ (𝜑 ↔ 𝜒) ∧ ((𝜑 → 𝜓) → (𝜓 ↔ 𝜃)))) & ⊢ 𝜑 & ⊢ 𝜓 & ⊢ 𝜒 & ⊢ 𝜃 ⇒ ⊢ 𝜏 | ||
Theorem | plvcofph 43231 | Given, a,b,d, and "definitions" for c, e, f: f is demonstrated. (Contributed by Jarvin Udandy, 8-Sep-2020.) |
⊢ (𝜒 ↔ ((((𝜑 ∧ 𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑)))) & ⊢ (𝜏 ↔ ((𝜒 → 𝜃) ∧ (𝜑 ↔ 𝜒) ∧ ((𝜑 → 𝜓) → (𝜓 ↔ 𝜃)))) & ⊢ (𝜂 ↔ (𝜒 ∧ 𝜏)) & ⊢ 𝜑 & ⊢ 𝜓 & ⊢ 𝜃 ⇒ ⊢ 𝜂 | ||
Theorem | plvcofphax 43232 | Given, a,b,d, and "definitions" for c, e, f, g: g is demonstrated. (Contributed by Jarvin Udandy, 8-Sep-2020.) |
⊢ (𝜒 ↔ ((((𝜑 ∧ 𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑)))) & ⊢ (𝜏 ↔ ((𝜒 → 𝜃) ∧ (𝜑 ↔ 𝜒) ∧ ((𝜑 → 𝜓) → (𝜓 ↔ 𝜃)))) & ⊢ (𝜂 ↔ (𝜒 ∧ 𝜏)) & ⊢ 𝜑 & ⊢ 𝜓 & ⊢ 𝜃 & ⊢ (𝜁 ↔ ¬ (𝜓 ∧ ¬ 𝜏)) ⇒ ⊢ 𝜁 | ||
Theorem | plvofpos 43233 | 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 ↔ (¬ 𝜑 ∧ 𝜓)) & ⊢ (𝜏 ↔ (𝜑 ∧ ¬ 𝜓)) & ⊢ (𝜂 ↔ (𝜑 ∧ 𝜓)) & ⊢ (𝜁 ↔ (((((¬ ((𝜇 → 𝜒) ∧ (𝜇 → 𝜃)) ∧ ¬ ((𝜇 → 𝜒) ∧ (𝜇 → 𝜏))) ∧ ¬ ((𝜇 → 𝜒) ∧ (𝜒 → 𝜂))) ∧ ¬ ((𝜇 → 𝜃) ∧ (𝜇 → 𝜏))) ∧ ¬ ((𝜇 → 𝜃) ∧ (𝜇 → 𝜂))) ∧ ¬ ((𝜇 → 𝜏) ∧ (𝜇 → 𝜂)))) & ⊢ (𝜎 ↔ (((𝜇 → 𝜒) ∨ (𝜇 → 𝜃)) ∨ ((𝜇 → 𝜏) ∨ (𝜇 → 𝜂)))) & ⊢ (𝜌 ↔ (𝜁 ∧ 𝜎)) & ⊢ 𝜁 & ⊢ 𝜎 ⇒ ⊢ 𝜌 | ||
Theorem | mdandyv0 43234 | 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 ↔ ⊥) ⇒ ⊢ ((((𝜒 ↔ 𝜑) ∧ (𝜃 ↔ 𝜑)) ∧ (𝜏 ↔ 𝜑)) ∧ (𝜂 ↔ 𝜑)) | ||
Theorem | mdandyv1 43235 | 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 ↔ ⊥) ⇒ ⊢ ((((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜑)) ∧ (𝜏 ↔ 𝜑)) ∧ (𝜂 ↔ 𝜑)) | ||
Theorem | mdandyv2 43236 | 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 ↔ ⊥) ⇒ ⊢ ((((𝜒 ↔ 𝜑) ∧ (𝜃 ↔ 𝜓)) ∧ (𝜏 ↔ 𝜑)) ∧ (𝜂 ↔ 𝜑)) | ||
Theorem | mdandyv3 43237 | 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 ↔ ⊥) ⇒ ⊢ ((((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜓)) ∧ (𝜏 ↔ 𝜑)) ∧ (𝜂 ↔ 𝜑)) | ||
Theorem | mdandyv4 43238 | 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 ↔ ⊥) ⇒ ⊢ ((((𝜒 ↔ 𝜑) ∧ (𝜃 ↔ 𝜑)) ∧ (𝜏 ↔ 𝜓)) ∧ (𝜂 ↔ 𝜑)) | ||
Theorem | mdandyv5 43239 | 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 ↔ ⊥) ⇒ ⊢ ((((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜑)) ∧ (𝜏 ↔ 𝜓)) ∧ (𝜂 ↔ 𝜑)) | ||
Theorem | mdandyv6 43240 | 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 ↔ ⊥) ⇒ ⊢ ((((𝜒 ↔ 𝜑) ∧ (𝜃 ↔ 𝜓)) ∧ (𝜏 ↔ 𝜓)) ∧ (𝜂 ↔ 𝜑)) | ||
Theorem | mdandyv7 43241 | 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 ↔ ⊥) ⇒ ⊢ ((((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜓)) ∧ (𝜏 ↔ 𝜓)) ∧ (𝜂 ↔ 𝜑)) | ||
Theorem | mdandyv8 43242 | 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 ↔ ⊤) ⇒ ⊢ ((((𝜒 ↔ 𝜑) ∧ (𝜃 ↔ 𝜑)) ∧ (𝜏 ↔ 𝜑)) ∧ (𝜂 ↔ 𝜓)) | ||
Theorem | mdandyv9 43243 | 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 ↔ ⊤) ⇒ ⊢ ((((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜑)) ∧ (𝜏 ↔ 𝜑)) ∧ (𝜂 ↔ 𝜓)) | ||
Theorem | mdandyv10 43244 | 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 ↔ ⊤) ⇒ ⊢ ((((𝜒 ↔ 𝜑) ∧ (𝜃 ↔ 𝜓)) ∧ (𝜏 ↔ 𝜑)) ∧ (𝜂 ↔ 𝜓)) | ||
Theorem | mdandyv11 43245 | 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 ↔ ⊤) ⇒ ⊢ ((((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜓)) ∧ (𝜏 ↔ 𝜑)) ∧ (𝜂 ↔ 𝜓)) | ||
Theorem | mdandyv12 43246 | 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 ↔ ⊤) ⇒ ⊢ ((((𝜒 ↔ 𝜑) ∧ (𝜃 ↔ 𝜑)) ∧ (𝜏 ↔ 𝜓)) ∧ (𝜂 ↔ 𝜓)) | ||
Theorem | mdandyv13 43247 | 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 ↔ ⊤) ⇒ ⊢ ((((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜑)) ∧ (𝜏 ↔ 𝜓)) ∧ (𝜂 ↔ 𝜓)) | ||
Theorem | mdandyv14 43248 | 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 ↔ ⊤) ⇒ ⊢ ((((𝜒 ↔ 𝜑) ∧ (𝜃 ↔ 𝜓)) ∧ (𝜏 ↔ 𝜓)) ∧ (𝜂 ↔ 𝜓)) | ||
Theorem | mdandyv15 43249 | 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 ↔ ⊤) ⇒ ⊢ ((((𝜒 ↔ 𝜓) ∧ (𝜃 ↔ 𝜓)) ∧ (𝜏 ↔ 𝜓)) ∧ (𝜂 ↔ 𝜓)) | ||
Theorem | mdandyvr0 43250 | 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.) |
⊢ (𝜑 ↔ 𝜁) & ⊢ (𝜓 ↔ 𝜎) & ⊢ (𝜒 ↔ 𝜑) & ⊢ (𝜃 ↔ 𝜑) & ⊢ (𝜏 ↔ 𝜑) & ⊢ (𝜂 ↔ 𝜑) ⇒ ⊢ ((((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜁)) ∧ (𝜏 ↔ 𝜁)) ∧ (𝜂 ↔ 𝜁)) | ||
Theorem | mdandyvr1 43251 | 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.) |
⊢ (𝜑 ↔ 𝜁) & ⊢ (𝜓 ↔ 𝜎) & ⊢ (𝜒 ↔ 𝜓) & ⊢ (𝜃 ↔ 𝜑) & ⊢ (𝜏 ↔ 𝜑) & ⊢ (𝜂 ↔ 𝜑) ⇒ ⊢ ((((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜁)) ∧ (𝜏 ↔ 𝜁)) ∧ (𝜂 ↔ 𝜁)) | ||
Theorem | mdandyvr2 43252 | 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.) |
⊢ (𝜑 ↔ 𝜁) & ⊢ (𝜓 ↔ 𝜎) & ⊢ (𝜒 ↔ 𝜑) & ⊢ (𝜃 ↔ 𝜓) & ⊢ (𝜏 ↔ 𝜑) & ⊢ (𝜂 ↔ 𝜑) ⇒ ⊢ ((((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜁)) ∧ (𝜂 ↔ 𝜁)) | ||
Theorem | mdandyvr3 43253 | 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.) |
⊢ (𝜑 ↔ 𝜁) & ⊢ (𝜓 ↔ 𝜎) & ⊢ (𝜒 ↔ 𝜓) & ⊢ (𝜃 ↔ 𝜓) & ⊢ (𝜏 ↔ 𝜑) & ⊢ (𝜂 ↔ 𝜑) ⇒ ⊢ ((((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜁)) ∧ (𝜂 ↔ 𝜁)) | ||
Theorem | mdandyvr4 43254 | 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.) |
⊢ (𝜑 ↔ 𝜁) & ⊢ (𝜓 ↔ 𝜎) & ⊢ (𝜒 ↔ 𝜑) & ⊢ (𝜃 ↔ 𝜑) & ⊢ (𝜏 ↔ 𝜓) & ⊢ (𝜂 ↔ 𝜑) ⇒ ⊢ ((((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜁)) ∧ (𝜏 ↔ 𝜎)) ∧ (𝜂 ↔ 𝜁)) | ||
Theorem | mdandyvr5 43255 | 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.) |
⊢ (𝜑 ↔ 𝜁) & ⊢ (𝜓 ↔ 𝜎) & ⊢ (𝜒 ↔ 𝜓) & ⊢ (𝜃 ↔ 𝜑) & ⊢ (𝜏 ↔ 𝜓) & ⊢ (𝜂 ↔ 𝜑) ⇒ ⊢ ((((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜁)) ∧ (𝜏 ↔ 𝜎)) ∧ (𝜂 ↔ 𝜁)) | ||
Theorem | mdandyvr6 43256 | 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.) |
⊢ (𝜑 ↔ 𝜁) & ⊢ (𝜓 ↔ 𝜎) & ⊢ (𝜒 ↔ 𝜑) & ⊢ (𝜃 ↔ 𝜓) & ⊢ (𝜏 ↔ 𝜓) & ⊢ (𝜂 ↔ 𝜑) ⇒ ⊢ ((((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜎)) ∧ (𝜂 ↔ 𝜁)) | ||
Theorem | mdandyvr7 43257 | 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.) |
⊢ (𝜑 ↔ 𝜁) & ⊢ (𝜓 ↔ 𝜎) & ⊢ (𝜒 ↔ 𝜓) & ⊢ (𝜃 ↔ 𝜓) & ⊢ (𝜏 ↔ 𝜓) & ⊢ (𝜂 ↔ 𝜑) ⇒ ⊢ ((((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜎)) ∧ (𝜂 ↔ 𝜁)) | ||
Theorem | mdandyvr8 43258 | 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.) |
⊢ (𝜑 ↔ 𝜁) & ⊢ (𝜓 ↔ 𝜎) & ⊢ (𝜒 ↔ 𝜑) & ⊢ (𝜃 ↔ 𝜑) & ⊢ (𝜏 ↔ 𝜑) & ⊢ (𝜂 ↔ 𝜓) ⇒ ⊢ ((((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜁)) ∧ (𝜏 ↔ 𝜁)) ∧ (𝜂 ↔ 𝜎)) | ||
Theorem | mdandyvr9 43259 | 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.) |
⊢ (𝜑 ↔ 𝜁) & ⊢ (𝜓 ↔ 𝜎) & ⊢ (𝜒 ↔ 𝜓) & ⊢ (𝜃 ↔ 𝜑) & ⊢ (𝜏 ↔ 𝜑) & ⊢ (𝜂 ↔ 𝜓) ⇒ ⊢ ((((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜁)) ∧ (𝜏 ↔ 𝜁)) ∧ (𝜂 ↔ 𝜎)) | ||
Theorem | mdandyvr10 43260 | 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.) |
⊢ (𝜑 ↔ 𝜁) & ⊢ (𝜓 ↔ 𝜎) & ⊢ (𝜒 ↔ 𝜑) & ⊢ (𝜃 ↔ 𝜓) & ⊢ (𝜏 ↔ 𝜑) & ⊢ (𝜂 ↔ 𝜓) ⇒ ⊢ ((((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜁)) ∧ (𝜂 ↔ 𝜎)) | ||
Theorem | mdandyvr11 43261 | 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.) |
⊢ (𝜑 ↔ 𝜁) & ⊢ (𝜓 ↔ 𝜎) & ⊢ (𝜒 ↔ 𝜓) & ⊢ (𝜃 ↔ 𝜓) & ⊢ (𝜏 ↔ 𝜑) & ⊢ (𝜂 ↔ 𝜓) ⇒ ⊢ ((((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜁)) ∧ (𝜂 ↔ 𝜎)) | ||
Theorem | mdandyvr12 43262 | 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.) |
⊢ (𝜑 ↔ 𝜁) & ⊢ (𝜓 ↔ 𝜎) & ⊢ (𝜒 ↔ 𝜑) & ⊢ (𝜃 ↔ 𝜑) & ⊢ (𝜏 ↔ 𝜓) & ⊢ (𝜂 ↔ 𝜓) ⇒ ⊢ ((((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜁)) ∧ (𝜏 ↔ 𝜎)) ∧ (𝜂 ↔ 𝜎)) | ||
Theorem | mdandyvr13 43263 | 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.) |
⊢ (𝜑 ↔ 𝜁) & ⊢ (𝜓 ↔ 𝜎) & ⊢ (𝜒 ↔ 𝜓) & ⊢ (𝜃 ↔ 𝜑) & ⊢ (𝜏 ↔ 𝜓) & ⊢ (𝜂 ↔ 𝜓) ⇒ ⊢ ((((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜁)) ∧ (𝜏 ↔ 𝜎)) ∧ (𝜂 ↔ 𝜎)) | ||
Theorem | mdandyvr14 43264 | 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.) |
⊢ (𝜑 ↔ 𝜁) & ⊢ (𝜓 ↔ 𝜎) & ⊢ (𝜒 ↔ 𝜑) & ⊢ (𝜃 ↔ 𝜓) & ⊢ (𝜏 ↔ 𝜓) & ⊢ (𝜂 ↔ 𝜓) ⇒ ⊢ ((((𝜒 ↔ 𝜁) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜎)) ∧ (𝜂 ↔ 𝜎)) | ||
Theorem | mdandyvr15 43265 | 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.) |
⊢ (𝜑 ↔ 𝜁) & ⊢ (𝜓 ↔ 𝜎) & ⊢ (𝜒 ↔ 𝜓) & ⊢ (𝜃 ↔ 𝜓) & ⊢ (𝜏 ↔ 𝜓) & ⊢ (𝜂 ↔ 𝜓) ⇒ ⊢ ((((𝜒 ↔ 𝜎) ∧ (𝜃 ↔ 𝜎)) ∧ (𝜏 ↔ 𝜎)) ∧ (𝜂 ↔ 𝜎)) | ||
Theorem | mdandyvrx0 43266 | 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.) |
⊢ (𝜑 ⊻ 𝜁) & ⊢ (𝜓 ⊻ 𝜎) & ⊢ (𝜒 ↔ 𝜑) & ⊢ (𝜃 ↔ 𝜑) & ⊢ (𝜏 ↔ 𝜑) & ⊢ (𝜂 ↔ 𝜑) ⇒ ⊢ ((((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜁)) ∧ (𝜏 ⊻ 𝜁)) ∧ (𝜂 ⊻ 𝜁)) | ||
Theorem | mdandyvrx1 43267 | 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.) |
⊢ (𝜑 ⊻ 𝜁) & ⊢ (𝜓 ⊻ 𝜎) & ⊢ (𝜒 ↔ 𝜓) & ⊢ (𝜃 ↔ 𝜑) & ⊢ (𝜏 ↔ 𝜑) & ⊢ (𝜂 ↔ 𝜑) ⇒ ⊢ ((((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜁)) ∧ (𝜏 ⊻ 𝜁)) ∧ (𝜂 ⊻ 𝜁)) | ||
Theorem | mdandyvrx2 43268 | 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.) |
⊢ (𝜑 ⊻ 𝜁) & ⊢ (𝜓 ⊻ 𝜎) & ⊢ (𝜒 ↔ 𝜑) & ⊢ (𝜃 ↔ 𝜓) & ⊢ (𝜏 ↔ 𝜑) & ⊢ (𝜂 ↔ 𝜑) ⇒ ⊢ ((((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜁)) ∧ (𝜂 ⊻ 𝜁)) | ||
Theorem | mdandyvrx3 43269 | 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.) |
⊢ (𝜑 ⊻ 𝜁) & ⊢ (𝜓 ⊻ 𝜎) & ⊢ (𝜒 ↔ 𝜓) & ⊢ (𝜃 ↔ 𝜓) & ⊢ (𝜏 ↔ 𝜑) & ⊢ (𝜂 ↔ 𝜑) ⇒ ⊢ ((((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜁)) ∧ (𝜂 ⊻ 𝜁)) | ||
Theorem | mdandyvrx4 43270 | 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.) |
⊢ (𝜑 ⊻ 𝜁) & ⊢ (𝜓 ⊻ 𝜎) & ⊢ (𝜒 ↔ 𝜑) & ⊢ (𝜃 ↔ 𝜑) & ⊢ (𝜏 ↔ 𝜓) & ⊢ (𝜂 ↔ 𝜑) ⇒ ⊢ ((((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜁)) ∧ (𝜏 ⊻ 𝜎)) ∧ (𝜂 ⊻ 𝜁)) | ||
Theorem | mdandyvrx5 43271 | 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.) |
⊢ (𝜑 ⊻ 𝜁) & ⊢ (𝜓 ⊻ 𝜎) & ⊢ (𝜒 ↔ 𝜓) & ⊢ (𝜃 ↔ 𝜑) & ⊢ (𝜏 ↔ 𝜓) & ⊢ (𝜂 ↔ 𝜑) ⇒ ⊢ ((((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜁)) ∧ (𝜏 ⊻ 𝜎)) ∧ (𝜂 ⊻ 𝜁)) | ||
Theorem | mdandyvrx6 43272 | 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.) |
⊢ (𝜑 ⊻ 𝜁) & ⊢ (𝜓 ⊻ 𝜎) & ⊢ (𝜒 ↔ 𝜑) & ⊢ (𝜃 ↔ 𝜓) & ⊢ (𝜏 ↔ 𝜓) & ⊢ (𝜂 ↔ 𝜑) ⇒ ⊢ ((((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜎)) ∧ (𝜂 ⊻ 𝜁)) | ||
Theorem | mdandyvrx7 43273 | 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.) |
⊢ (𝜑 ⊻ 𝜁) & ⊢ (𝜓 ⊻ 𝜎) & ⊢ (𝜒 ↔ 𝜓) & ⊢ (𝜃 ↔ 𝜓) & ⊢ (𝜏 ↔ 𝜓) & ⊢ (𝜂 ↔ 𝜑) ⇒ ⊢ ((((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜎)) ∧ (𝜂 ⊻ 𝜁)) | ||
Theorem | mdandyvrx8 43274 | 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.) |
⊢ (𝜑 ⊻ 𝜁) & ⊢ (𝜓 ⊻ 𝜎) & ⊢ (𝜒 ↔ 𝜑) & ⊢ (𝜃 ↔ 𝜑) & ⊢ (𝜏 ↔ 𝜑) & ⊢ (𝜂 ↔ 𝜓) ⇒ ⊢ ((((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜁)) ∧ (𝜏 ⊻ 𝜁)) ∧ (𝜂 ⊻ 𝜎)) | ||
Theorem | mdandyvrx9 43275 | 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.) |
⊢ (𝜑 ⊻ 𝜁) & ⊢ (𝜓 ⊻ 𝜎) & ⊢ (𝜒 ↔ 𝜓) & ⊢ (𝜃 ↔ 𝜑) & ⊢ (𝜏 ↔ 𝜑) & ⊢ (𝜂 ↔ 𝜓) ⇒ ⊢ ((((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜁)) ∧ (𝜏 ⊻ 𝜁)) ∧ (𝜂 ⊻ 𝜎)) | ||
Theorem | mdandyvrx10 43276 | 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.) |
⊢ (𝜑 ⊻ 𝜁) & ⊢ (𝜓 ⊻ 𝜎) & ⊢ (𝜒 ↔ 𝜑) & ⊢ (𝜃 ↔ 𝜓) & ⊢ (𝜏 ↔ 𝜑) & ⊢ (𝜂 ↔ 𝜓) ⇒ ⊢ ((((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜁)) ∧ (𝜂 ⊻ 𝜎)) | ||
Theorem | mdandyvrx11 43277 | 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.) |
⊢ (𝜑 ⊻ 𝜁) & ⊢ (𝜓 ⊻ 𝜎) & ⊢ (𝜒 ↔ 𝜓) & ⊢ (𝜃 ↔ 𝜓) & ⊢ (𝜏 ↔ 𝜑) & ⊢ (𝜂 ↔ 𝜓) ⇒ ⊢ ((((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜁)) ∧ (𝜂 ⊻ 𝜎)) | ||
Theorem | mdandyvrx12 43278 | 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.) |
⊢ (𝜑 ⊻ 𝜁) & ⊢ (𝜓 ⊻ 𝜎) & ⊢ (𝜒 ↔ 𝜑) & ⊢ (𝜃 ↔ 𝜑) & ⊢ (𝜏 ↔ 𝜓) & ⊢ (𝜂 ↔ 𝜓) ⇒ ⊢ ((((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜁)) ∧ (𝜏 ⊻ 𝜎)) ∧ (𝜂 ⊻ 𝜎)) | ||
Theorem | mdandyvrx13 43279 | 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.) |
⊢ (𝜑 ⊻ 𝜁) & ⊢ (𝜓 ⊻ 𝜎) & ⊢ (𝜒 ↔ 𝜓) & ⊢ (𝜃 ↔ 𝜑) & ⊢ (𝜏 ↔ 𝜓) & ⊢ (𝜂 ↔ 𝜓) ⇒ ⊢ ((((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜁)) ∧ (𝜏 ⊻ 𝜎)) ∧ (𝜂 ⊻ 𝜎)) | ||
Theorem | mdandyvrx14 43280 | 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.) |
⊢ (𝜑 ⊻ 𝜁) & ⊢ (𝜓 ⊻ 𝜎) & ⊢ (𝜒 ↔ 𝜑) & ⊢ (𝜃 ↔ 𝜓) & ⊢ (𝜏 ↔ 𝜓) & ⊢ (𝜂 ↔ 𝜓) ⇒ ⊢ ((((𝜒 ⊻ 𝜁) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜎)) ∧ (𝜂 ⊻ 𝜎)) | ||
Theorem | mdandyvrx15 43281 | 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.) |
⊢ (𝜑 ⊻ 𝜁) & ⊢ (𝜓 ⊻ 𝜎) & ⊢ (𝜒 ↔ 𝜓) & ⊢ (𝜃 ↔ 𝜓) & ⊢ (𝜏 ↔ 𝜓) & ⊢ (𝜂 ↔ 𝜓) ⇒ ⊢ ((((𝜒 ⊻ 𝜎) ∧ (𝜃 ⊻ 𝜎)) ∧ (𝜏 ⊻ 𝜎)) ∧ (𝜂 ⊻ 𝜎)) | ||
Theorem | H15NH16TH15IH16 43282 | 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 ⇒ ⊢ (((((((((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) ∧ jph) ∧ jps) ∧ jch) → jth) | ||
Theorem | dandysum2p2e4 43283 |
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 ↔ (𝜃 ∧ 𝜏)) & ⊢ (𝜓 ↔ (𝜂 ∧ 𝜁)) & ⊢ (𝜒 ↔ (𝜎 ∧ 𝜌)) & ⊢ (𝜃 ↔ ⊥) & ⊢ (𝜏 ↔ ⊥) & ⊢ (𝜂 ↔ ⊤) & ⊢ (𝜁 ↔ ⊤) & ⊢ (𝜎 ↔ ⊥) & ⊢ (𝜌 ↔ ⊥) & ⊢ (𝜇 ↔ ⊥) & ⊢ (𝜆 ↔ ⊥) & ⊢ (𝜅 ↔ ((𝜃 ⊻ 𝜏) ⊻ (𝜃 ∧ 𝜏))) & ⊢ (jph ↔ ((𝜂 ⊻ 𝜁) ∨ 𝜑)) & ⊢ (jps ↔ ((𝜎 ⊻ 𝜌) ∨ 𝜓)) & ⊢ (jch ↔ ((𝜇 ⊻ 𝜆) ∨ 𝜒)) ⇒ ⊢ ((((((((((((((((𝜑 ↔ (𝜃 ∧ 𝜏)) ∧ (𝜓 ↔ (𝜂 ∧ 𝜁))) ∧ (𝜒 ↔ (𝜎 ∧ 𝜌))) ∧ (𝜃 ↔ ⊥)) ∧ (𝜏 ↔ ⊥)) ∧ (𝜂 ↔ ⊤)) ∧ (𝜁 ↔ ⊤)) ∧ (𝜎 ↔ ⊥)) ∧ (𝜌 ↔ ⊥)) ∧ (𝜇 ↔ ⊥)) ∧ (𝜆 ↔ ⊥)) ∧ (𝜅 ↔ ((𝜃 ⊻ 𝜏) ⊻ (𝜃 ∧ 𝜏)))) ∧ (jph ↔ ((𝜂 ⊻ 𝜁) ∨ 𝜑))) ∧ (jps ↔ ((𝜎 ⊻ 𝜌) ∨ 𝜓))) ∧ (jch ↔ ((𝜇 ⊻ 𝜆) ∨ 𝜒))) → ((((𝜅 ↔ ⊥) ∧ (jph ↔ ⊥)) ∧ (jps ↔ ⊤)) ∧ (jch ↔ ⊥))) | ||
Theorem | mdandysum2p2e4 43284 |
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 ↔ (𝜃 ∧ 𝜏)) & ⊢ (𝜓 ↔ (𝜂 ∧ 𝜁)) & ⊢ (𝜒 ↔ (𝜎 ∧ 𝜌)) & ⊢ (𝜃 ↔ jth) & ⊢ (𝜏 ↔ jth) & ⊢ (𝜂 ↔ jta) & ⊢ (𝜁 ↔ jta) & ⊢ (𝜎 ↔ jth) & ⊢ (𝜌 ↔ jth) & ⊢ (𝜇 ↔ jth) & ⊢ (𝜆 ↔ jth) & ⊢ (𝜅 ↔ ((𝜃 ⊻ 𝜏) ⊻ (𝜃 ∧ 𝜏))) & ⊢ (jph ↔ ((𝜂 ⊻ 𝜁) ∨ 𝜑)) & ⊢ (jps ↔ ((𝜎 ⊻ 𝜌) ∨ 𝜓)) & ⊢ (jch ↔ ((𝜇 ⊻ 𝜆) ∨ 𝜒)) ⇒ ⊢ ((((((((((((((((𝜑 ↔ (𝜃 ∧ 𝜏)) ∧ (𝜓 ↔ (𝜂 ∧ 𝜁))) ∧ (𝜒 ↔ (𝜎 ∧ 𝜌))) ∧ (𝜃 ↔ ⊥)) ∧ (𝜏 ↔ ⊥)) ∧ (𝜂 ↔ ⊤)) ∧ (𝜁 ↔ ⊤)) ∧ (𝜎 ↔ ⊥)) ∧ (𝜌 ↔ ⊥)) ∧ (𝜇 ↔ ⊥)) ∧ (𝜆 ↔ ⊥)) ∧ (𝜅 ↔ ((𝜃 ⊻ 𝜏) ⊻ (𝜃 ∧ 𝜏)))) ∧ (jph ↔ ((𝜂 ⊻ 𝜁) ∨ 𝜑))) ∧ (jps ↔ ((𝜎 ⊻ 𝜌) ∨ 𝜓))) ∧ (jch ↔ ((𝜇 ⊻ 𝜆) ∨ 𝜒))) → ((((𝜅 ↔ ⊥) ∧ (jph ↔ ⊥)) ∧ (jps ↔ ⊤)) ∧ (jch ↔ ⊥))) | ||
Theorem | adh-jarrsc 43285 | 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.) |
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜃 → (𝜓 → 𝜒))) | ||
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 43286, or with the single axiom adh-minimp 43298. This section proves first adh-minim 43286 from { ax-1 6, ax-2 7 }, followed by the converse, due to Ivo Thomas; and then it proves adh-minimp 43298 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 43286 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 43286 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.) | ||
Theorem | adh-minim 43286 | 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.) |
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜃 → ((𝜓 → (𝜒 → 𝜏)) → (𝜓 → 𝜏)))) | ||
Theorem | adh-minim-ax1-ax2-lem1 43287 | First lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 43286 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 → (𝜓 → 𝜏)) → (𝜃 → 𝜏))) → 𝜂)) → (𝜓 → 𝜂))) | ||
Theorem | adh-minim-ax1-ax2-lem2 43288 | Second lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 43286 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 → (𝜑 → 𝜃)) → (𝜒 → 𝜃))) → 𝜏)) → (𝜑 → 𝜏)) | ||
Theorem | adh-minim-ax1-ax2-lem3 43289 | Third lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 43286 and ax-mp 5. Polish prefix notation: CCpCqrCqCsCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜃 → (𝜑 → 𝜒)))) | ||
Theorem | adh-minim-ax1-ax2-lem4 43290 | Fourth lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 43286 and ax-mp 5. Polish prefix notation: CCCpqrCCqCrsCqs . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜓 → (𝜒 → 𝜃)) → (𝜓 → 𝜃))) | ||
Theorem | adh-minim-ax1 43291 | Derivation of ax-1 6 from adh-minim 43286 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 → (𝜓 → 𝜑)) | ||
Theorem | adh-minim-ax2-lem5 43292 | Fifth lemma for the derivation of ax-2 7 from adh-minim 43286 and ax-mp 5. Polish prefix notation: CpCCCqrsCCrCstCrt . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (((𝜓 → 𝜒) → 𝜃) → ((𝜒 → (𝜃 → 𝜏)) → (𝜒 → 𝜏)))) | ||
Theorem | adh-minim-ax2-lem6 43293 | Sixth lemma for the derivation of ax-2 7 from adh-minim 43286 and ax-mp 5. Polish prefix notation: CCpCCCCqrsCCrCstCrtuCpu . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → ((((𝜓 → 𝜒) → 𝜃) → ((𝜒 → (𝜃 → 𝜏)) → (𝜒 → 𝜏))) → 𝜂)) → (𝜑 → 𝜂)) | ||
Theorem | adh-minim-ax2c 43294 | Derivation of a commuted form of ax-2 7 from adh-minim 43286 and ax-mp 5. Polish prefix notation: CCpqCCpCqrCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) | ||
Theorem | adh-minim-ax2 43295 | Derivation of ax-2 7 from adh-minim 43286 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 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | ||
Theorem | adh-minim-idALT 43296 | Derivation of id 22 (reflexivity of implication, PM *2.08 WhiteheadRussell p. 101) from adh-minim-ax1 43291, adh-minim-ax2 43295, 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.) |
⊢ (𝜑 → 𝜑) | ||
Theorem | adh-minim-pm2.43 43297 | Derivation of pm2.43 56 WhiteheadRussell p. 106 (also called "hilbert" or "W") from adh-minim-ax1 43291, adh-minim-ax2 43295, 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 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)) | ||
Theorem | adh-minimp 43298 | 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 → ((𝜓 → 𝜒) → (((𝜃 → 𝜓) → (𝜒 → 𝜏)) → (𝜓 → 𝜏)))) | ||
Theorem | adh-minimp-jarr-imim1-ax2c-lem1 43299 | 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 43298 and ax-mp 5. Polish prefix notation: CCpqCCCrpCqsCps . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → (((𝜒 → 𝜑) → (𝜓 → 𝜃)) → (𝜑 → 𝜃))) | ||
Theorem | adh-minimp-jarr-lem2 43300 | 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 43298 and ax-mp 5. Polish prefix notation: CCCpqCCCrsCCCtrCsuCruvCqv . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → (((𝜒 → 𝜃) → (((𝜏 → 𝜒) → (𝜃 → 𝜂)) → (𝜒 → 𝜂))) → 𝜁)) → (𝜓 → 𝜁)) |
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