P. 448 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44701-44800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eelT00 44701	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 44702	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 44703	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 44704	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 44705	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 44706	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 44707	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 44708	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 44709	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 44710	syl2an 596 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 44711	syl2an 596 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 44712	el0321old 44713 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 44713	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 44714	el2122old 44715 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 44715	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 44716	Elimination rule similar to mp4an 693, 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 44717	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 44718	Elimination rule similar eel0000 44716, 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 44719	Five-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl113anc 1384 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 44720	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 44721	Conjunction form of e12 44720 (see syl6an 684). (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 44722	Virtual deduction form of syl2an 596. (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 44723	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 44724	Conjunction form of e20 44723. (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 44725	e20an 44724 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 44726	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 44727	Conjunction form of e21 44726. (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 44728	e21an 44727 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 44729	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 44730	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 44731	Conjunction form of e33 44730. (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 44732	e33an 44731 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 44733	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 44734	Biconditional form of e3 44733. syl8ib 256 is e3bi 44734 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 44735	Right biconditional form of e3 44733. (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 44736	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 44737	e03 44736 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 44738	Conjunction form of e03 44736. (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 44739	Conjunction form of ee03 44737. (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 44740	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 44741	e30 44740 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 44742	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 44743	Conjunction form of ee30 44741. (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 44744	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 44745	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 44746	e13an 44745 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 44747	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 44748	e31 44747 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 44749	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 44750	e31an 44749 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 44751	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 44752	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 44753	e23an 44752 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 44754	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 44755	e32 44754 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 44756	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 44757	e33an 44731 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 44758	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 44759	e123 44758 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 44760	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 44761	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 44762	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 44763	A virtual deduction elimination rule. The non-virtual deduction form of e000 44763 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 44764	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 44765	Elimination rule identical to mp2an 692. The non-virtual deduction form is the virtual deduction form, which is mp2an 692. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ 𝜒
Theorem	eel00cT 44766	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 44767	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 44768	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 44769	An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⊤ → 𝜑)    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ 𝜓
Theorem	eel0cT 44770	An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (⊤ → 𝜓)
Theorem	eelT0 44771	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 44772	Elimination rule identical to mpbi 230. The non-virtual deduction form is the virtual deduction form, which is mpbi 230. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ 𝜓
Theorem	e0bir 44773	Elimination rule identical to mpbir 231. The non-virtual deduction form is the virtual deduction form, which is mpbir 231. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ (𝜓 ↔ 𝜑)    ⇒   ⊢ 𝜓
Theorem	uun0.1 44774	Convention notation form of un0.1 44775. (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 44775	⊤ is the constant true, a tautology (see df-tru 1543). 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 44776	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 1543. (Revised by David A. Wheeler, 8-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((⊤ ∧ 𝜑) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	uunT1p1 44777	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 44778	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 44779	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 44780	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 44781	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 44782	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 44783	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 673. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜓 ∧ 𝜑) ∧ 𝜑) → 𝜒)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → 𝜒)
Theorem	un10 44784	A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   (   𝜑   ,   ⊤   )   ▶   𝜓   )    ⇒   ⊢ (   𝜑   ▶   𝜓   )
Theorem	un01 44785	A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (   (   ⊤   ,   𝜑   )   ▶   𝜓   )    ⇒   ⊢ (   𝜑   ▶   𝜓   )
Theorem	un2122 44786	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.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	uun2131 44787	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)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	uun2131p1 44788	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 ∧ 𝜒) ∧ (𝜑 ∧ 𝜓)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
Theorem	uunTT1 44789	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	uunTT1p1 44790	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	uunTT1p2 44791	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	uunT11 44792	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)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	uunT11p1 44793	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)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	uunT11p2 44794	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)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	uunT12 44795	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)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	uunT12p1 44796	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 ∧ 𝜑) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	uunT12p2 44797	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)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	uunT12p3 44798	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 ∧ ⊤ ∧ 𝜑) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	uunT12p4 44799	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)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	uunT12p5 44800	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 ∧ 𝜑 ∧ ⊤) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)