Home Metamath Proof ExplorerTheorem List (p. 388 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27745) Hilbert Space Explorer (27746-29270) Users' Mathboxes (29271-42316)

Theorem List for Metamath Proof Explorer - 38701-38800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoreme121 38701 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   )    &   (   𝜑   ▶   𝜏   )    &   (𝜓 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜒   ▶   𝜂   )

Theoreme211 38702 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   )    &   (   𝜑   ▶   𝜏   )    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )

Theoremee211 38703 e211 38702 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))

Theoreme210 38704 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   )    &   𝜏    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )

Theoremee210 38705 e210 38704 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)    &   𝜏    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))

Theoreme201 38706 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   )    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )

Theoremee201 38707 e201 38706 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   𝜃    &   (𝜑𝜏)    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))

Theoreme120 38708 A virtual deduction elimination rule. (Contributed by Alan Sare, 10-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ,   𝜒   ▶   𝜃   )    &   𝜏    &   (𝜓 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜒   ▶   𝜂   )

Theoremee120 38709 Virtual deduction rule e120 38708 without virtual deduction symbols. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))    &   𝜏    &   (𝜓 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜒𝜂))

Theoreme021 38710 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   ▶   𝜏   )    &   (𝜑 → (𝜃 → (𝜏𝜂)))       (   𝜓   ,   𝜒   ▶   𝜂   )

Theoremee021 38711 e021 38710 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓 → (𝜒𝜃))    &   (𝜓𝜏)    &   (𝜑 → (𝜃 → (𝜏𝜂)))       (𝜓 → (𝜒𝜂))

Theoreme012 38712 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   ▶   𝜏   )    &   (𝜑 → (𝜒 → (𝜏𝜂)))       (   𝜓   ,   𝜃   ▶   𝜂   )

Theoremee012 38713 e012 38712 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓𝜒)    &   (𝜓 → (𝜃𝜏))    &   (𝜑 → (𝜒 → (𝜏𝜂)))       (𝜓 → (𝜃𝜂))

Theoreme102 38714 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   ▶   𝜏   )    &   (𝜓 → (𝜒 → (𝜏𝜂)))       (   𝜑   ,   𝜃   ▶   𝜂   )

Theoremee102 38715 e102 38714 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   𝜒    &   (𝜑 → (𝜃𝜏))    &   (𝜓 → (𝜒 → (𝜏𝜂)))       (𝜑 → (𝜃𝜂))

Theoreme22 38716 A virtual deduction elimination rule. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (   𝜑   ,   𝜓   ▶   𝜃   )    &   (𝜒 → (𝜃𝜏))       (   𝜑   ,   𝜓   ▶   𝜏   )

Theoreme22an 38717 Conjunction form of e22 38716. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (   𝜑   ,   𝜓   ▶   𝜃   )    &   ((𝜒𝜃) → 𝜏)       (   𝜑   ,   𝜓   ▶   𝜏   )

Theoremee22an 38718 e22an 38717 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   ((𝜒𝜃) → 𝜏)       (𝜑 → (𝜓𝜏))

Theoreme111 38719 A virtual deduction elimination rule (see syl3c 66). (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ▶   𝜒   )    &   (   𝜑   ▶   𝜃   )    &   (𝜓 → (𝜒 → (𝜃𝜏)))       (   𝜑   ▶   𝜏   )

Theoreme1111 38720 A virtual deduction elimination rule. (Contributed by Alan Sare, 6-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ▶   𝜒   )    &   (   𝜑   ▶   𝜃   )    &   (   𝜑   ▶   𝜏   )    &   (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂))))       (   𝜑   ▶   𝜂   )

Theoreme110 38721 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    &   (𝜓 → (𝜒 → (𝜃𝜏)))       (   𝜑   ▶   𝜏   )

Theoremee110 38722 e110 38721 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑𝜒)    &   𝜃    &   (𝜓 → (𝜒 → (𝜃𝜏)))       (𝜑𝜏)

Theoreme101 38723 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 → (𝜃𝜏)))       (   𝜑   ▶   𝜏   )

Theoremee101 38724 e101 38723 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   𝜒    &   (𝜑𝜃)    &   (𝜓 → (𝜒 → (𝜃𝜏)))       (𝜑𝜏)

Theoreme011 38725 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 → (𝜃𝜏)))       (   𝜓   ▶   𝜏   )

Theoremee011 38726 e011 38725 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓𝜒)    &   (𝜓𝜃)    &   (𝜑 → (𝜒 → (𝜃𝜏)))       (𝜓𝜏)

Theoreme100 38727 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 → (𝜃𝜏)))       (   𝜑   ▶   𝜏   )

Theoremee100 38728 e100 38727 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   𝜒    &   𝜃    &   (𝜓 → (𝜒 → (𝜃𝜏)))       (𝜑𝜏)

Theoreme010 38729 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 → (𝜃𝜏)))       (   𝜓   ▶   𝜏   )

Theoremee010 38730 e010 38729 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓𝜒)    &   𝜃    &   (𝜑 → (𝜒 → (𝜃𝜏)))       (𝜓𝜏)

Theoreme001 38731 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 → (𝜃𝜏)))       (   𝜒   ▶   𝜏   )

Theoremee001 38732 e001 38731 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   (𝜒𝜃)    &   (𝜑 → (𝜓 → (𝜃𝜏)))       (𝜒𝜏)

Theoreme11 38733 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 → (𝜒𝜃))       (   𝜑   ▶   𝜃   )

Theoreme11an 38734 Conjunction form of e11 38733. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ▶   𝜒   )    &   ((𝜓𝜒) → 𝜃)       (   𝜑   ▶   𝜃   )

Theoremee11an 38735 e11an 38734 without virtual deductions. syl22anc 1325 is also e11an 38734 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.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)

Theoreme01 38736 A virtual deduction elimination rule. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   𝜓   ▶   𝜒   )    &   (𝜑 → (𝜒𝜃))       (   𝜓   ▶   𝜃   )

Theoreme01an 38737 Conjunction form of e01 38736. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   𝜓   ▶   𝜒   )    &   ((𝜑𝜒) → 𝜃)       (   𝜓   ▶   𝜃   )

Theoremee01an 38738 e01an 38737 without virtual deductions. sylancr 694 is also a form of e01an 38737 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    &   (𝜓𝜒)    &   ((𝜑𝜒) → 𝜃)       (𝜓𝜃)

Theoreme10 38739 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 → (𝜒𝜃))       (   𝜑   ▶   𝜃   )

Theoreme10an 38740 Conjunction form of e10 38739. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   𝜒    &   ((𝜓𝜒) → 𝜃)       (   𝜑   ▶   𝜃   )

Theoremee10an 38741 e10an 38740 without virtual deductions. sylancl 693 is also e10an 38740 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.)
(𝜑𝜓)    &   𝜒    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)

Theoreme02 38742 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 → (𝜃𝜏))       (   𝜓   ,   𝜒   ▶   𝜏   )

Theoreme02an 38743 Conjunction form of e02 38742. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   𝜓   ,   𝜒   ▶   𝜃   )    &   ((𝜑𝜃) → 𝜏)       (   𝜓   ,   𝜒   ▶   𝜏   )

Theoremee02an 38744 e02an 38743 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓 → (𝜒𝜃))    &   ((𝜑𝜃) → 𝜏)       (𝜓 → (𝜒𝜏))

Theoremeel021old 38745 el021old 38746 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   ((𝜓𝜒) → 𝜃)    &   ((𝜑𝜃) → 𝜏)       ((𝜓𝜒) → 𝜏)

Theoremel021old 38746 A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   (   𝜓   ,   𝜒   )   ▶   𝜃   )    &   ((𝜑𝜃) → 𝜏)       (   (   𝜓   ,   𝜒   )   ▶   𝜏   )

Theoremeel132 38747 syl2an 494 with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016.)
(𝜑𝜓)    &   ((𝜒𝜃) → 𝜏)    &   ((𝜓𝜏) → 𝜂)       ((𝜑𝜒𝜃) → 𝜂)

Theoremeel000cT 38748 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       (⊤ → 𝜃)

Theoremeel0TT 38749 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (⊤ → 𝜓)    &   (⊤ → 𝜒)    &   ((𝜑𝜓𝜒) → 𝜃)       𝜃

TheoremeelT00 38750 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   𝜓    &   𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       𝜃

TheoremeelTTT 38751 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (⊤ → 𝜓)    &   (⊤ → 𝜒)    &   ((𝜑𝜓𝜒) → 𝜃)       𝜃

TheoremeelT11 38752 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (𝜓𝜒)    &   (𝜓𝜃)    &   ((𝜑𝜒𝜃) → 𝜏)       (𝜓𝜏)

TheoremeelT1 38753 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)    &   (𝜓𝜒)    &   ((𝜑𝜒) → 𝜃)       (𝜓𝜃)

TheoremeelT12 38754 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (𝜓𝜒)    &   (𝜃𝜏)    &   ((𝜑𝜒𝜏) → 𝜂)       ((𝜓𝜃) → 𝜂)

TheoremeelTT1 38755 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (⊤ → 𝜓)    &   (𝜒𝜃)    &   ((𝜑𝜓𝜃) → 𝜏)       (𝜒𝜏)

TheoremeelT01 38756 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   𝜓    &   (𝜒𝜃)    &   ((𝜑𝜓𝜃) → 𝜏)       (𝜒𝜏)

Theoremeel0T1 38757 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (⊤ → 𝜓)    &   (𝜒𝜃)    &   ((𝜑𝜓𝜃) → 𝜏)       (𝜒𝜏)

Theoremeel12131 38758 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
(𝜑𝜓)    &   ((𝜑𝜒) → 𝜃)    &   ((𝜑𝜏) → 𝜂)    &   ((𝜓𝜃𝜂) → 𝜁)       ((𝜑𝜒𝜏) → 𝜁)

Theoremeel2131 38759 syl2an 494 with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜏)    &   ((𝜒𝜏) → 𝜂)       ((𝜑𝜓𝜃) → 𝜂)

Theoremeel3132 38760 syl2an 494 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
((𝜑𝜓) → 𝜒)    &   ((𝜃𝜓) → 𝜏)    &   ((𝜒𝜏) → 𝜂)       ((𝜑𝜃𝜓) → 𝜂)

Theoremeel0321old 38761 el0321old 38762 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   ((𝜓𝜒𝜃) → 𝜏)    &   ((𝜑𝜏) → 𝜂)       ((𝜓𝜒𝜃) → 𝜂)

Theoremel0321old 38762 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   )    &   ((𝜑𝜏) → 𝜂)       (   (   𝜓   ,   𝜒   ,   𝜃   )   ▶   𝜂   )

Theoremeel2122old 38763 el2122old 38764 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → 𝜒)    &   (𝜓𝜃)    &   (𝜓𝜏)    &   ((𝜒𝜃𝜏) → 𝜂)       ((𝜑𝜓) → 𝜂)

Theoremel2122old 38764 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   )    &   ((𝜒𝜃𝜏) → 𝜂)       (   (   𝜑   ,   𝜓   )   ▶   𝜂   )

Theoremeel0001 38765 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
𝜑    &   𝜓    &   𝜒    &   (𝜃𝜏)    &   ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜂)       (𝜃𝜂)

Theoremeel0000 38766 Elimination rule similar to mp4an 708, except with a left-nested conjunction unification theorem. (Contributed by Alan Sare, 17-Oct-2017.)
𝜑    &   𝜓    &   𝜒    &   𝜃    &   ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       𝜏

Theoremeel1111 38767 Four-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl112anc 1328 except the unification theorem uses left-nested conjunction. (Contributed by Alan Sare, 17-Oct-2017.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   ((((𝜓𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)       (𝜑𝜂)

Theoremeel00001 38768 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
𝜑    &   𝜓    &   𝜒    &   𝜃    &   (𝜏𝜂)    &   (((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜂) → 𝜁)       (𝜏𝜁)

Theoremeel00000 38769 Elimination rule similar eel0000 38766, except with five hpothesis steps. (Contributed by Alan Sare, 17-Oct-2017.)
𝜑    &   𝜓    &   𝜒    &   𝜃    &   𝜏    &   (((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)       𝜂

Theoremeel11111 38770 Five-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl113anc 1336 except the unification theorem uses left-nested conjunction. (Contributed by Alan Sare, 17-Oct-2017.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((((𝜓𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜁)       (𝜑𝜁)

Theoreme12 38771 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 → (𝜃𝜏))       (   𝜑   ,   𝜒   ▶   𝜏   )

Theoreme12an 38772 Conjunction form of e12 38771 (see syl6an 567). (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ,   𝜒   ▶   𝜃   )    &   ((𝜓𝜃) → 𝜏)       (   𝜑   ,   𝜒   ▶   𝜏   )

Theoremel12 38773 Virtual deduction form of syl2an 494. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜏   ▶   𝜒   )    &   ((𝜓𝜒) → 𝜃)       (   (   𝜑   ,   𝜏   )   ▶   𝜃   )

Theoreme20 38774 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 → (𝜃𝜏))       (   𝜑   ,   𝜓   ▶   𝜏   )

Theoreme20an 38775 Conjunction form of e20 38774. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   𝜃    &   ((𝜒𝜃) → 𝜏)       (   𝜑   ,   𝜓   ▶   𝜏   )

Theoremee20an 38776 e20an 38775 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   𝜃    &   ((𝜒𝜃) → 𝜏)       (𝜑 → (𝜓𝜏))

Theoreme21 38777 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 → (𝜃𝜏))       (   𝜑   ,   𝜓   ▶   𝜏   )

Theoreme21an 38778 Conjunction form of e21 38777. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (   𝜑   ▶   𝜃   )    &   ((𝜒𝜃) → 𝜏)       (   𝜑   ,   𝜓   ▶   𝜏   )

Theoremee21an 38779 e21an 38778 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)    &   ((𝜒𝜃) → 𝜏)       (𝜑 → (𝜓𝜏))

Theoreme333 38780 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 → (𝜂𝜁)))       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜁   )

Theoreme33 38781 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 → (𝜏𝜂))       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Theoreme33an 38782 Conjunction form of e33 38781. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )    &   ((𝜃𝜏) → 𝜂)       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Theoremee33an 38783 e33an 38782 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜒𝜏)))    &   ((𝜃𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒𝜂)))

Theoreme3 38784 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   )    &   (𝜃𝜏)       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )

Theoreme3bi 38785 Biconditional form of e3 38784. syl8ib 246 is e3bi 38785 without virtual deductions. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   (𝜃𝜏)       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )

Theoreme3bir 38786 Right biconditional form of e3 38784. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   (𝜏𝜃)       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜏   )

Theoreme03 38787 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 → (𝜏𝜂))       (   𝜓   ,   𝜒   ,   𝜃   ▶   𝜂   )

Theoremee03 38788 e03 38787 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓 → (𝜒 → (𝜃𝜏)))    &   (𝜑 → (𝜏𝜂))       (𝜓 → (𝜒 → (𝜃𝜂)))

Theoreme03an 38789 Conjunction form of e03 38787. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   𝜓   ,   𝜒   ,   𝜃   ▶   𝜏   )    &   ((𝜑𝜏) → 𝜂)       (   𝜓   ,   𝜒   ,   𝜃   ▶   𝜂   )

Theoremee03an 38790 Conjunction form of ee03 38788. (Contributed by Alan Sare, 18-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓 → (𝜒 → (𝜃𝜏)))    &   ((𝜑𝜏) → 𝜂)       (𝜓 → (𝜒 → (𝜃𝜂)))

Theoreme30 38791 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 → (𝜏𝜂))       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Theoremee30 38792 e30 38791 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   𝜏    &   (𝜃 → (𝜏𝜂))       (𝜑 → (𝜓 → (𝜒𝜂)))

Theoreme30an 38793 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    &   ((𝜃𝜏) → 𝜂)       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Theoremee30an 38794 Conjunction form of ee30 38792. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   𝜏    &   ((𝜃𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒𝜂)))

Theoreme13 38795 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 → (𝜏𝜂))       (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜂   )

Theoreme13an 38796 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   )    &   ((𝜓𝜏) → 𝜂)       (   𝜑   ,   𝜒   ,   𝜃   ▶   𝜂   )

Theoremee13an 38797 e13an 38796 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → (𝜒 → (𝜃𝜏)))    &   ((𝜓𝜏) → 𝜂)       (𝜑 → (𝜒 → (𝜃𝜂)))

Theoreme31 38798 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 → (𝜏𝜂))       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Theoremee31 38799 e31 38798 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑𝜏)    &   (𝜃 → (𝜏𝜂))       (𝜑 → (𝜓 → (𝜒𝜂)))

Theoreme31an 38800 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   )    &   ((𝜃𝜏) → 𝜂)       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
 Copyright terms: Public domain < Previous  Next >