P. 482 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48101-48200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cliftetb 48101	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 48102	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 48103	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 48104	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 48105	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 48106	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 48107	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 48108	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 48109	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 48110	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 48111	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 48112	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 48113	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 48114	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 48115	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 48116	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 48117	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 48118	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 48119	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 48120	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 48121	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 48122	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 48123	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 48124	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 48125	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 48126	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 48127	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 48128	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 48129	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 48130	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 48131	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 48132	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 48133	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 48134	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 48135	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 48136	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 48137	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 48138	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 48139	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 48140	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 48141	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 48142	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 48143	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 48144	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 48145	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 48146	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 48147	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 48148	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 48149	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 48150	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 48151	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 48152	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 48153	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 48154	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 48155	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 48156	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 48157	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 48158	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 48159	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 48160	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 48161	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 would 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 48162	CONTRADICTION PROVED AT 1 + 1 = 2 . Luckily Mario Carneiro did a successful version of his own. See Mario's Relevant Work: 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 would 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 ↔ ⊥)))
21.51  Mathbox for Adhemar
Theorem	adh-jarrsc 48163	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)))
21.51.1  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 84, ax-1 6, pm2.43 57 } (written B'KW), or with { imim2 59, pm2.04 91, ax-1 6, pm2.43 57 } (written BCKW), or with the single axiom adh-minim 48164, or with the single axiom adh-minimp 48176. This section proves first adh-minim 48164 from { ax-1 6, ax-2 7 }, followed by the converse, due to Ivo Thomas; and then it proves adh-minimp 48176 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 48164 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 48164 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 48164	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 48165	First lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 48164 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 48166	Second lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 48164 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 48167	Third lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 48164 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 48168	Fourth lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 48164 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 48169	Derivation of ax-1 6 from adh-minim 48164 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 48170	Fifth lemma for the derivation of ax-2 7 from adh-minim 48164 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 48171	Sixth lemma for the derivation of ax-2 7 from adh-minim 48164 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 48172	Derivation of a commuted form of ax-2 7 from adh-minim 48164 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 48173	Derivation of ax-2 7 from adh-minim 48164 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 48174	Derivation of id 23 (reflexivity of implication, PM *2.08 WhiteheadRussell p. 101) from adh-minim-ax1 48169, adh-minim-ax2 48173, 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 48175	Derivation of pm2.43 57 WhiteheadRussell p. 106 (also called "hilbert" or "W") from adh-minim-ax1 48169, adh-minim-ax2 48173, 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 48176	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 48177	First lemma for the derivation of jarr 107, imim1 84, and a commuted form of ax-2 7, and indirectly ax-1 6 and ax-2 7, from adh-minimp 48176 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 48178	Second lemma for the derivation of jarr 107, and indirectly ax-1 6, a commuted form of ax-2 7, and ax-2 7 proper, from adh-minimp 48176 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))
Theorem	adh-minimp-jarr-ax2c-lem3 48179	Third lemma for the derivation of jarr 107 and a commuted form of ax-2 7, and indirectly ax-1 6 and ax-2 7 proper , from adh-minimp 48176 and ax-mp 5. Polish prefix notation: CCCCpqCCCrpCqsCpstt . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((𝜑 → 𝜓) → (((𝜒 → 𝜑) → (𝜓 → 𝜃)) → (𝜑 → 𝜃))) → 𝜏) → 𝜏)
Theorem	adh-minimp-sylsimp 48180	Derivation of jarr 107 (also called "syll-simp") from minimp 1653 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.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))
Theorem	adh-minimp-ax1 48181	Derivation of ax-1 6 from adh-minimp 48176 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 → (𝜓 → 𝜑))
Theorem	adh-minimp-imim1 48182	Derivation of imim1 84 ("left antimonotonicity of implication", theorem *2.06 of [WhiteheadRussell] p. 100) from adh-minimp 48176 and ax-mp 5. Polish prefix notation: CCpqCCqrCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
Theorem	adh-minimp-ax2c 48183	Derivation of a commuted form of ax-2 7 from adh-minimp 48176 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.)
⊢ ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))
Theorem	adh-minimp-ax2-lem4 48184	Fourth lemma for the derivation of ax-2 7 from adh-minimp 48176 and ax-mp 5. Polish prefix notation: CpCCqCprCqr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → ((𝜓 → (𝜑 → 𝜒)) → (𝜓 → 𝜒)))
Theorem	adh-minimp-ax2 48185	Derivation of ax-2 7 from adh-minimp 48176 and ax-mp 5. Polish prefix notation: CCpCqrCCpqCpr . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
Theorem	adh-minimp-idALT 48186	Derivation of id 23 (reflexivity of implication, PM *2.08 WhiteheadRussell p. 101) from adh-minimp-ax1 48181, adh-minimp-ax2 48185, 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-minimp-pm2.43 48187	Derivation of pm2.43 57 WhiteheadRussell p. 106 (also called "hilbert" or "W") from adh-minimp-ax1 48181, adh-minimp-ax2 48185, and ax-mp 5. It uses the derivation written DD22D21 in D-notation. (See head comment for an explanation.) Polish prefix notation: CCpCpqCpq . (Contributed by BJ, 31-May-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))
21.52  Mathbox for Alexander van der Vekens
21.52.1  General auxiliary theorems (1)
21.52.1.1  Unordered and ordered pairs - extension for singletons
Theorem	n0nsn2el 48188*	If a class with one element is not a singleton, there is at least another element in this class. (Contributed by AV, 6-Mar-2025.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≠ {𝐴}) → ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)
Theorem	eusnsn 48189*	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 48190*	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 48191*	Another way to express existential uniqueness of a wff 𝜑: its associated class abstraction {𝑥 ∣ 𝜑} is a singleton. Variant of euabsn2 4682 using existential uniqueness for the singleton element instead of existence only. (Contributed by AV, 24-Aug-2022.)
⊢ (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥 ∣ 𝜑} = {𝑦})
21.52.1.2  Unordered and ordered pairs - extension for unordered pairs
Theorem	elprneb 48192	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))
21.52.1.3  Unordered and ordered pairs - extension for ordered pairs
Theorem	oppr 48193	Equality for ordered pairs implies equality of unordered pairs with the same elements. (Contributed by AV, 9-Jul-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴, 𝐵} = {𝐶, 𝐷}))
Theorem	opprb 48194	Equality for unordered pairs corresponds to equality of unordered pairs with the same elements. (Contributed by AV, 9-Jul-2023.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ∨ ⟨𝐴, 𝐵⟩ = ⟨𝐷, 𝐶⟩)))
Theorem	or2expropbilem1 48195*	Lemma 1 for or2expropbi 48197 and ich2exprop 48646. (Contributed by AV, 16-Jul-2023.)
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 = 𝑎 ∧ 𝐵 = 𝑏) → (𝜑 → ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
Theorem	or2expropbilem2 48196*	Lemma 2 for or2expropbi 48197 and ich2exprop 48646. (Contributed by AV, 16-Jul-2023.)
⊢ (∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))
Theorem	or2expropbi 48197*	If two classes are strictly ordered, there is an ordered pair of both classes fulfilling a wff iff there is an unordered pair of both classes fulfilling the wff. (Contributed by AV, 26-Aug-2023.)
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑅 Or 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴𝑅𝐵)) → (∃𝑎∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ (𝑎𝑅𝑏 ∧ 𝜑)) ↔ ∃𝑎∃𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝑅𝑏 ∧ 𝜑))))
21.52.1.4  Relations - extension
Theorem	eubrv 48198*	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 48199*	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)
Theorem	eldmressn 48200	Element of the domain of a restriction to a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
⊢ (𝐵 ∈ dom (𝐹 ↾ {𝐴}) → 𝐵 = 𝐴)