P. 456 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 45501-45600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	e101 45501	A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ 𝜒    &   ⊢ (   𝜑   ▶   𝜃   )    &   ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))    ⇒   ⊢ (   𝜑   ▶   𝜏   )
Theorem	ee101 45502	e101 45501 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    &   ⊢ 𝜒    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))    ⇒   ⊢ (𝜑 → 𝜏)
Theorem	e011 45503	A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (   𝜓   ▶   𝜒   )    &   ⊢ (   𝜓   ▶   𝜃   )    &   ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜏)))    ⇒   ⊢ (   𝜓   ▶   𝜏   )
Theorem	ee011 45504	e011 45503 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜓 → 𝜒)    &   ⊢ (𝜓 → 𝜃)    &   ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜏)))    ⇒   ⊢ (𝜓 → 𝜏)
Theorem	e100 45505	A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   𝜑   ▶   𝜓   )    &   ⊢ 𝜒    &   ⊢ 𝜃    &   ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))    ⇒   ⊢ (   𝜑   ▶   𝜏   )
Theorem	ee100 45506	e100 45505 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝜓)    &   ⊢ 𝜒    &   ⊢ 𝜃    &   ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜏)))    ⇒   ⊢ (𝜑 → 𝜏)
Theorem	e010 45507	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 → (𝜒 → (𝜃 → 𝜏)))    ⇒   ⊢ (   𝜓   ▶   𝜏   )
Theorem	ee010 45508	e010 45507 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜓 → 𝜒)    &   ⊢ 𝜃    &   ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜏)))    ⇒   ⊢ (𝜓 → 𝜏)
Theorem	e001 45509	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 → (𝜃 → 𝜏)))    ⇒   ⊢ (   𝜒   ▶   𝜏   )
Theorem	ee001 45510	e001 45509 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 45511	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 45512	Conjunction form of e11 45511. (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 45513	e11an 45512 without virtual deductions. syl22anc 852 is also e11an 45512 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 45514	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 45515	Conjunction form of e01 45514. (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 45516	e01an 45515 without virtual deductions. sylancr 599 is also a form of e01an 45515 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 45517	A virtual deduction elimination rule (see mpisyl 22). (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 45518	Conjunction form of e10 45517. (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 45519	e10an 45518 without virtual deductions. sylancl 598 is also e10an 45518 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 45520	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 45521	Conjunction form of e02 45520. (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 45522	e02an 45521 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 45523	el021old 45524 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 45524	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 45525	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 45526	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 45527	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 45528	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 45529	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 45530	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 45531	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 45532	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 45533	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 45534	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 45535	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 45536	syl2an 608 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 45537	syl2an 608 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 45538	el0321old 45539 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 45539	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 45540	el2122old 45541 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 45541	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 45542	Elimination rule similar to mp4an 706, 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 45543	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 45544	Elimination rule similar eel0000 45542, 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 45545	Five-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl113anc 1409 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 45546	A virtual deduction elimination rule (see sylsyld 62). (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 45547	Conjunction form of e12 45546 (see syl6an 697). (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 45548	Virtual deduction form of syl2an 608. (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 45549	A virtual deduction elimination rule (see syl6mpi 68). (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 45550	Conjunction form of e20 45549. (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 45551	e20an 45550 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 45552	A virtual deduction elimination rule (see syl6ci 72). (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 45553	Conjunction form of e21 45552. (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 45554	e21an 45553 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 45555	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 45556	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 45557	Conjunction form of e33 45556. (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 45558	e33an 45557 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 45559	Meta-connective form of syl8 77. (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 45560	Biconditional form of e3 45559. syl8ib 259 is e3bi 45560 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 45561	Right biconditional form of e3 45559. (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 45562	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 45563	e03 45562 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 45564	Conjunction form of e03 45562. (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 45565	Conjunction form of ee03 45563. (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 45566	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 45567	e30 45566 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 45568	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 45569	Conjunction form of ee30 45567. (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 45570	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 45571	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 45572	e13an 45571 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 45573	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 45574	e31 45573 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 45575	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 45576	e31an 45575 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 45577	A virtual deduction elimination rule (see syl10 80). (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 45578	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 45579	e23an 45578 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 45580	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 45581	e32 45580 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 45582	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 45583	e33an 45557 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 45584	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 45585	e123 45584 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 45586	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 45587	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 45588	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 45589	A virtual deduction elimination rule. The non-virtual deduction form of e000 45589 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 45590	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 45591	Elimination rule identical to mp2an 705. The non-virtual deduction form is the virtual deduction form, which is mp2an 705. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ 𝜒
Theorem	eel00cT 45592	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 45593	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 45594	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 45595	An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ 𝜓
Theorem	eel0cT 45596	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (⊤ → 𝜓)
Theorem	eelT0 45597	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 45598	Elimination rule identical to mpbi 233. The non-virtual deduction form is the virtual deduction form, which is mpbi 233. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ 𝜓
Theorem	e0bir 45599	Elimination rule identical to mpbir 234. The non-virtual deduction form is the virtual deduction form, which is mpbir 234. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜓 ↔ 𝜑)    ⇒   ⊢ 𝜓
Theorem	uun0.1 45600	Convention notation form of un0.1 45601. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ (𝜓 → 𝜒)    &   ⊢ ((⊤ ∧ 𝜓) → 𝜃)    ⇒   ⊢ (𝜓 → 𝜃)