![]() |
Metamath
Proof Explorer Theorem List (p. 421 of 437) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-28347) |
![]() (28348-29872) |
![]() (29873-43657) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | aibnbaif 42001 | Given a implies b, not b, there exists a proof for a is F. (Contributed by Jarvin Udandy, 1-Sep-2016.) |
⊢ (𝜑 → 𝜓) & ⊢ ¬ 𝜓 ⇒ ⊢ (𝜑 ↔ ⊥) | ||
Theorem | aiffbtbat 42002 | Given a is equivalent to b, T. is equivalent to b. there exists a proof for a is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (⊤ ↔ 𝜓) ⇒ ⊢ (𝜑 ↔ ⊤) | ||
Theorem | astbstanbst 42003 | Given a is equivalent to T., also given that b is equivalent to T, there exists a proof for a and b is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.) |
⊢ (𝜑 ↔ ⊤) & ⊢ (𝜓 ↔ ⊤) ⇒ ⊢ ((𝜑 ∧ 𝜓) ↔ ⊤) | ||
Theorem | aistbistaandb 42004 | Given a is equivalent to T., also given that b is equivalent to T, there exists a proof for (a and b). (Contributed by Jarvin Udandy, 9-Sep-2016.) |
⊢ (𝜑 ↔ ⊤) & ⊢ (𝜓 ↔ ⊤) ⇒ ⊢ (𝜑 ∧ 𝜓) | ||
Theorem | aisbnaxb 42005 | Given a is equivalent to b, there exists a proof for (not (a xor b)). (Contributed by Jarvin Udandy, 28-Aug-2016.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ¬ (𝜑 ⊻ 𝜓) | ||
Theorem | atbiffatnnb 42006 | If a implies b, then a implies not not b. (Contributed by Jarvin Udandy, 28-Aug-2016.) |
⊢ ((𝜑 → 𝜓) → (𝜑 → ¬ ¬ 𝜓)) | ||
Theorem | bisaiaisb 42007 | Application of bicom1 with a, b swapped. (Contributed by Jarvin Udandy, 31-Aug-2016.) |
⊢ ((𝜓 ↔ 𝜑) → (𝜑 ↔ 𝜓)) | ||
Theorem | atbiffatnnbalt 42008 | If a implies b, then a implies not not b. (Contributed by Jarvin Udandy, 29-Aug-2016.) |
⊢ ((𝜑 → 𝜓) → (𝜑 → ¬ ¬ 𝜓)) | ||
Theorem | abnotbtaxb 42009 | Assuming a, not b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.) |
⊢ 𝜑 & ⊢ ¬ 𝜓 ⇒ ⊢ (𝜑 ⊻ 𝜓) | ||
Theorem | abnotataxb 42010 | Assuming not a, b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.) |
⊢ ¬ 𝜑 & ⊢ 𝜓 ⇒ ⊢ (𝜑 ⊻ 𝜓) | ||
Theorem | conimpf 42011 | 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 42012 | 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 42013 | 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 42014 | 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 42015 | 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 42016 | 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 42017 | 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 42018 | 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 42019 | 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 42020 | 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 42021 | 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 42022 | 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 42023 | 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 42024 | 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 42025 | 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 42026 | For when pm3.4 lacks a pm3.4i. (Contributed by Jarvin Udandy, 9-Sep-2020.) |
⊢ (𝜑 ∧ 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | onenotinotbothi 42027 | 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 42028 | 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 42029 | 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 42030 | 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 42031 | 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 42032 | 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 42033 | 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 42034 | 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 42035 | 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 42036 | 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 42037 | 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 42038 | 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 42039 | 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 42040 | 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 42041 | 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 42042 | 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 42043 | 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 42044 | 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 42045 | 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 42046 | 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 42047 | 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 42048 | 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 42049 | 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 42050 | 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 42051 | 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 42052 | 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 42053 | 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 42054 | 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 42055 | 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 42056 | 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 42057 | 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 42058 | 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 42059 | 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 42060 | 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 42061 | 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 42062 | 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 42063 | 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 42064 | 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 42065 | 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 42066 | 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 42067 | 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 42068 | 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 42069 | 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 42070 | 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 42071 | 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 42072 | 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 42073 | 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 42074 | 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 42075 | 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 42076 | 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 42077 | 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 42078 | 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 42079 | 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 42080 | 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 42081 | 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 42082 | 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 42083 | 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 42084 | 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 42085 | 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 42086 | 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 42087 | 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 42088 | 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 42089 | 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 42090 | 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 42091 | 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 42092 |
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 42093 |
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 | raaan2 42094* | Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 4303. It is necessary that either both restricting classes are empty or both are not empty. (Contributed by Alexander van der Vekens, 29-Jun-2017.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ ((𝐴 = ∅ ↔ 𝐵 = ∅) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))) | ||
Theorem | eusnsn 42095* | There is a unique element of a singleton which is equal to another singleton. (Contributed by AV, 24-Aug-2022.) |
⊢ ∃!𝑥{𝑥} = {𝑦} | ||
Theorem | absnsb 42096* | If the class abstraction {𝑥 ∣ 𝜑} associated with the wff 𝜑 is a singleton, the wff is true for the singleton element. (Contributed by AV, 24-Aug-2022.) |
⊢ ({𝑥 ∣ 𝜑} = {𝑦} → [𝑦 / 𝑥]𝜑) | ||
Theorem | euabsneu 42097* | Another way to express existential uniqueness of a wff 𝜑: its associated class abstraction {𝑥 ∣ 𝜑} is a singleton. Variant of euabsn2 4492 using existential uniqueness for the singleton element instead of existence only. (Contributed by AV, 24-Aug-2022.) |
⊢ (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥 ∣ 𝜑} = {𝑦}) | ||
Theorem | elprneb 42098 | An element of a proper unordered pair is the first element iff it is not the second element. (Contributed by AV, 18-Jun-2020.) |
⊢ ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐵 ≠ 𝐶) → (𝐴 = 𝐵 ↔ 𝐴 ≠ 𝐶)) | ||
Theorem | eubrv 42099* | If there is a unique set which is related to a class, then the class must be a set. (Contributed by AV, 25-Aug-2022.) |
⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ V) | ||
Theorem | eubrdm 42100* | If there is a unique set which is related to a class, then the class is an element of the domain of the relation. (Contributed by AV, 25-Aug-2022.) |
⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ dom 𝑅) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |