P. 453 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 45201-45300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ee001 45201	e001 45200 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ (𝜒 → 𝜃)    &   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))    ⇒   ⊢ (𝜒 → 𝜏)
Theorem	e11 45202	A virtual deduction elimination rule. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ (   𝜑   ▶   𝜒   )    &   ⊢ (𝜓 → (𝜒 → 𝜃))    ⇒   ⊢ (   𝜑   ▶   𝜃   )
Theorem	e11an 45203	Conjunction form of e11 45202. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ (   𝜑   ▶   𝜒   )    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (   𝜑   ▶   𝜃   )
Theorem	ee11an 45204	e11an 45203 without virtual deductions. syl22anc 847 is also e11an 45203 without virtual deductions, exept with a different order of hypotheses. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	e01 45205	A virtual deduction elimination rule. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (   𝜓   ▶   𝜒   )    &   ⊢ (𝜑 → (𝜒 → 𝜃))    ⇒   ⊢ (   𝜓   ▶   𝜃   )
Theorem	e01an 45206	Conjunction form of e01 45205. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (   𝜓   ▶   𝜒   )    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (   𝜓   ▶   𝜃   )
Theorem	ee01an 45207	e01an 45206 without virtual deductions. sylancr 595 is also a form of e01an 45206 without virtual deduction, except the order of the hypotheses is different. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜓 → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜓 → 𝜃)
Theorem	e10 45208	A virtual deduction elimination rule (see mpisyl 21). (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ 𝜒    &   ⊢ (𝜓 → (𝜒 → 𝜃))    ⇒   ⊢ (   𝜑   ▶   𝜃   )
Theorem	e10an 45209	Conjunction form of e10 45208. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ 𝜒    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (   𝜑   ▶   𝜃   )
Theorem	ee10an 45210	e10an 45209 without virtual deductions. sylancl 594 is also e10an 45209 without virtual deductions, except the order of the hypotheses is different. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    &   ⊢ 𝜒    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	e02 45211	A virtual deduction elimination rule. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ (𝜑 → (𝜃 → 𝜏))    ⇒   ⊢ (   𝜓   ,   𝜒   ▶   𝜏   )
Theorem	e02an 45212	Conjunction form of e02 45211. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (   𝜓   ,   𝜒   ▶   𝜏   )
Theorem	ee02an 45213	e02an 45212 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜓 → (𝜒 → 𝜃))    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜓 → (𝜒 → 𝜏))
Theorem	eel021old 45214	el021old 45215 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜓 ∧ 𝜒) → 𝜏)
Theorem	el021old 45215	A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (   (   𝜓   ,   𝜒   )   ▶   𝜃   )    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (   (   𝜓   ,   𝜒   )   ▶   𝜏   )
Theorem	eel000cT 45216	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (⊤ → 𝜃)
Theorem	eel0TT 45217	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (⊤ → 𝜓)    &   ⊢ (⊤ → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ 𝜃
Theorem	eelT00 45218	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ 𝜃
Theorem	eelTTT 45219	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ (⊤ → 𝜓)    &   ⊢ (⊤ → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ 𝜃
Theorem	eelT11 45220	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ (𝜓 → 𝜒)    &   ⊢ (𝜓 → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜓 → 𝜏)
Theorem	eelT1 45221	Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Alan Sare, 23-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ (𝜓 → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜓 → 𝜃)
Theorem	eelT12 45222	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ (𝜓 → 𝜒)    &   ⊢ (𝜃 → 𝜏)    &   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)    ⇒   ⊢ ((𝜓 ∧ 𝜃) → 𝜂)
Theorem	eelTT1 45223	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ (⊤ → 𝜓)    &   ⊢ (𝜒 → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜒 → 𝜏)
Theorem	eelT01 45224	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ 𝜓    &   ⊢ (𝜒 → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜒 → 𝜏)
Theorem	eel0T1 45225	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (⊤ → 𝜓)    &   ⊢ (𝜒 → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜒 → 𝜏)
Theorem	eel12131 45226	An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ (𝜑 → 𝜓)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜏) → 𝜂)    &   ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)    ⇒   ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁)
Theorem	eel2131 45227	syl2an 604 with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜏)    &   ⊢ ((𝜒 ∧ 𝜏) → 𝜂)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜂)
Theorem	eel3132 45228	syl2an 604 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜃 ∧ 𝜓) → 𝜏)    &   ⊢ ((𝜒 ∧ 𝜏) → 𝜂)    ⇒   ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜓) → 𝜂)
Theorem	eel0321old 45229	el0321old 45230 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)    &   ⊢ ((𝜑 ∧ 𝜏) → 𝜂)    ⇒   ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜂)
Theorem	el0321old 45230	A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (   (   𝜓   ,   𝜒   ,   𝜃   )   ▶   𝜏   )    &   ⊢ ((𝜑 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (   (   𝜓   ,   𝜒   ,   𝜃   )   ▶   𝜂   )
Theorem	eel2122old 45231	el2122old 45232 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ (𝜓 → 𝜃)    &   ⊢ (𝜓 → 𝜏)    &   ⊢ ((𝜒 ∧ 𝜃 ∧ 𝜏) → 𝜂)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜂)
Theorem	el2122old 45232	A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   (   𝜑   ,   𝜓   )   ▶   𝜒   )    &   ⊢ (   𝜓   ▶   𝜃   )    &   ⊢ (   𝜓   ▶   𝜏   )    &   ⊢ ((𝜒 ∧ 𝜃 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (   (   𝜑   ,   𝜓   )   ▶   𝜂   )
Theorem	eel0000 45233	Elimination rule similar to mp4an 701, except with a left-nested conjunction unification theorem. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ 𝜃    &   ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ 𝜏
Theorem	eel00001 45234	An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ 𝜃    &   ⊢ (𝜏 → 𝜂)    &   ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜂) → 𝜁)    ⇒   ⊢ (𝜏 → 𝜁)
Theorem	eel00000 45235	Elimination rule similar eel0000 45233, except with five hpothesis steps. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ 𝜃    &   ⊢ 𝜏    &   ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)    ⇒   ⊢ 𝜂
Theorem	eel11111 45236	Five-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl113anc 1393 except the unification theorem uses left-nested conjunction. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜑 → 𝜏)    &   ⊢ (𝜑 → 𝜂)    &   ⊢ (((((𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜁)    ⇒   ⊢ (𝜑 → 𝜁)
Theorem	e12 45237	A virtual deduction elimination rule (see sylsyld 61). (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ (   𝜑   ,   𝜒   ▶   𝜃   )    &   ⊢ (𝜓 → (𝜃 → 𝜏))    ⇒   ⊢ (   𝜑   ,   𝜒   ▶   𝜏   )
Theorem	e12an 45238	Conjunction form of e12 45237 (see syl6an 692). (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ (   𝜑   ,   𝜒   ▶   𝜃   )    &   ⊢ ((𝜓 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (   𝜑   ,   𝜒   ▶   𝜏   )
Theorem	el12 45239	Virtual deduction form of syl2an 604. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ (   𝜏   ▶   𝜒   )    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (   (   𝜑   ,   𝜏   )   ▶   𝜃   )
Theorem	e20 45240	A virtual deduction elimination rule (see syl6mpi 67). (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ▶   𝜒   )    &   ⊢ 𝜃    &   ⊢ (𝜒 → (𝜃 → 𝜏))    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )
Theorem	e20an 45241	Conjunction form of e20 45240. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ▶   𝜒   )    &   ⊢ 𝜃    &   ⊢ ((𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )
Theorem	ee20an 45242	e20an 45241 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ 𝜃    &   ⊢ ((𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜏))
Theorem	e21 45243	A virtual deduction elimination rule (see syl6ci 71). (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ▶   𝜒   )    &   ⊢ (   𝜑   ▶   𝜃   )    &   ⊢ (𝜒 → (𝜃 → 𝜏))    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )
Theorem	e21an 45244	Conjunction form of e21 45243. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ▶   𝜒   )    &   ⊢ (   𝜑   ▶   𝜃   )    &   ⊢ ((𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )
Theorem	ee21an 45245	e21an 45244 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → 𝜃)    &   ⊢ ((𝜒 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜏))
Theorem	e333 45246	A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )    &   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )    &   ⊢ (𝜃 → (𝜏 → (𝜂 → 𝜁)))    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜁   )
Theorem	e33 45247	A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )    &   ⊢ (𝜃 → (𝜏 → 𝜂))    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
Theorem	e33an 45248	Conjunction form of e33 45247. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )    &   ⊢ ((𝜃 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
Theorem	ee33an 45249	e33an 45248 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))    &   ⊢ ((𝜃 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))
Theorem	e3 45250	Meta-connective form of syl8 76. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ (𝜃 → 𝜏)    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )
Theorem	e3bi 45251	Biconditional form of e3 45250. syl8ib 258 is e3bi 45251 without virtual deductions. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ (𝜃 ↔ 𝜏)    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )
Theorem	e3bir 45252	Right biconditional form of e3 45250. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ (𝜏 ↔ 𝜃)    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )
Theorem	e03 45253	A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (   𝜓   ,   𝜒   ,   𝜃   ▶   𝜏   )    &   ⊢ (𝜑 → (𝜏 → 𝜂))    ⇒   ⊢ (   𝜓   ,   𝜒   ,   𝜃   ▶   𝜂   )
Theorem	ee03 45254	e03 45253 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))    &   ⊢ (𝜑 → (𝜏 → 𝜂))    ⇒   ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜂)))
Theorem	e03an 45255	Conjunction form of e03 45253. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (   𝜓   ,   𝜒   ,   𝜃   ▶   𝜏   )    &   ⊢ ((𝜑 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (   𝜓   ,   𝜒   ,   𝜃   ▶   𝜂   )
Theorem	ee03an 45256	Conjunction form of ee03 45254. (Contributed by Alan Sare, 18-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))    &   ⊢ ((𝜑 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜂)))
Theorem	e30 45257	A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ 𝜏    &   ⊢ (𝜃 → (𝜏 → 𝜂))    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
Theorem	ee30 45258	e30 45257 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ 𝜏    &   ⊢ (𝜃 → (𝜏 → 𝜂))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))
Theorem	e30an 45259	A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ 𝜏    &   ⊢ ((𝜃 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
Theorem	ee30an 45260	Conjunction form of ee30 45258. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ 𝜏    &   ⊢ ((𝜃 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))
Theorem	e13 45261	A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜏   )    &   ⊢ (𝜓 → (𝜏 → 𝜂))    ⇒   ⊢ (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜂   )
Theorem	e13an 45262	A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜏   )    &   ⊢ ((𝜓 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜂   )
Theorem	ee13an 45263	e13an 45262 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜏)))    &   ⊢ ((𝜓 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜂)))
Theorem	e31 45264	A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ (   𝜑   ▶   𝜏   )    &   ⊢ (𝜃 → (𝜏 → 𝜂))    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
Theorem	ee31 45265	e31 45264 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜑 → 𝜏)    &   ⊢ (𝜃 → (𝜏 → 𝜂))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))
Theorem	e31an 45266	A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ (   𝜑   ▶   𝜏   )    &   ⊢ ((𝜃 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
Theorem	ee31an 45267	e31an 45266 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜑 → 𝜏)    &   ⊢ ((𝜃 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))
Theorem	e23 45268	A virtual deduction elimination rule (see syl10 79). (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ▶   𝜒   )    &   ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜏   )    &   ⊢ (𝜒 → (𝜏 → 𝜂))    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )
Theorem	e23an 45269	A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ▶   𝜒   )    &   ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜏   )    &   ⊢ ((𝜒 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )
Theorem	ee23an 45270	e23an 45269 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))    &   ⊢ ((𝜒 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂)))
Theorem	e32 45271	A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )    &   ⊢ (𝜃 → (𝜏 → 𝜂))    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
Theorem	ee32 45272	e32 45271 without virtual deductions. (Contributed by Alan Sare, 18-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜑 → (𝜓 → 𝜏))    &   ⊢ (𝜃 → (𝜏 → 𝜂))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))
Theorem	e32an 45273	A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )    &   ⊢ ((𝜃 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
Theorem	ee32an 45274	e33an 45248 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜑 → (𝜓 → 𝜏))    &   ⊢ ((𝜃 ∧ 𝜏) → 𝜂)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))
Theorem	e123 45275	A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ (   𝜑   ,   𝜒   ▶   𝜃   )    &   ⊢ (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜂   )    &   ⊢ (𝜓 → (𝜃 → (𝜂 → 𝜁)))    ⇒   ⊢ (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜁   )
Theorem	ee123 45276	e123 45275 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜒 → 𝜃))    &   ⊢ (𝜑 → (𝜒 → (𝜏 → 𝜂)))    &   ⊢ (𝜓 → (𝜃 → (𝜂 → 𝜁)))    ⇒   ⊢ (𝜑 → (𝜒 → (𝜏 → 𝜁)))
Theorem	el123 45277	A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ (   𝜒   ▶   𝜃   )    &   ⊢ (   𝜏   ▶   𝜂   )    &   ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)    ⇒   ⊢ (   (   𝜑   ,   𝜒   ,   𝜏   )   ▶   𝜁   )
Theorem	e233 45278	A virtual deduction elimination rule. (Contributed by Alan Sare, 29-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ▶   𝜒   )    &   ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜏   )    &   ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )    &   ⊢ (𝜒 → (𝜏 → (𝜂 → 𝜁)))    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜁   )
Theorem	e323 45279	A virtual deduction elimination rule. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   ⊢ (   𝜑   ,   𝜓   ▶   𝜏   )    &   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )    &   ⊢ (𝜃 → (𝜏 → (𝜂 → 𝜁)))    ⇒   ⊢ (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜁   )
Theorem	e000 45280	A virtual deduction elimination rule. The non-virtual deduction form of e000 45280 is the virtual deduction form. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    ⇒   ⊢ 𝜃
Theorem	e00 45281	Elimination rule identical to mp2 9. The non-virtual deduction form is the virtual deduction form, which is mp2 9. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ 𝜒
Theorem	e00an 45282	Elimination rule identical to mp2an 700. The non-virtual deduction form is the virtual deduction form, which is mp2an 700. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ 𝜒
Theorem	eel00cT 45283	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (⊤ → 𝜒)
Theorem	eelTT 45284	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ (⊤ → 𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ 𝜒
Theorem	e0a 45285	Elimination rule identical to ax-mp 5. The non-virtual deduction form is the virtual deduction form, which is ax-mp 5. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ 𝜓
Theorem	eelT 45286	An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ 𝜓
Theorem	eel0cT 45287	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (⊤ → 𝜓)
Theorem	eelT0 45288	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ 𝜓    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ 𝜒
Theorem	e0bi 45289	Elimination rule identical to mpbi 232. The non-virtual deduction form is the virtual deduction form, which is mpbi 232. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ 𝜓
Theorem	e0bir 45290	Elimination rule identical to mpbir 233. The non-virtual deduction form is the virtual deduction form, which is mpbir 233. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜓 ↔ 𝜑)    ⇒   ⊢ 𝜓
Theorem	uun0.1 45291	Convention notation form of un0.1 45292. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ (𝜓 → 𝜒)    &   ⊢ ((⊤ ∧ 𝜓) → 𝜃)    ⇒   ⊢ (𝜓 → 𝜃)
Theorem	un0.1 45292	⊤ is the constant true, a tautology (see df-tru 1553). Kleene's "empty conjunction" is logically equivalent to ⊤. In a virtual deduction we shall interpret ⊤ to be the empty wff or the empty collection of virtual hypotheses. ⊤ in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   ⊤   ▶   𝜑   )    &   ⊢ (   𝜓   ▶   𝜒   )    &   ⊢ (   (   ⊤   ,   𝜓   )   ▶   𝜃   )    ⇒   ⊢ (   𝜓   ▶   𝜃   )
Theorem	uunT1 45293	A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) Proof was revised to accommodate a possible future version of df-tru 1553. (Revised by David A. Wheeler, 8-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((⊤ ∧ 𝜑) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	uunT1p1 45294	A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ ⊤) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	uunT21 45295	A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((⊤ ∧ (𝜑 ∧ 𝜓)) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	uun121 45296	A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	uun121p1 45297	A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜑) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	uun132 45298	A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	uun132p1 45299	A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜓 ∧ 𝜒) ∧ 𝜑) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	anabss7p1 45300	A deduction unionizing a non-unionized collection of virtual hypotheses. This would have been named uun221 if the zeroth permutation did not exist in set.mm as anabss7 681. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜓 ∧ 𝜑) ∧ 𝜑) → 𝜒)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → 𝜒)