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Theorem List for Metamath Proof Explorer - 40901-41000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremee001 40901 e001 40900 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 40902 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 40903 Conjunction form of e11 40902. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ▶   𝜒   )    &   ((𝜓𝜒) → 𝜃)       (   𝜑   ▶   𝜃   )
 
Theoremee11an 40904 e11an 40903 without virtual deductions. syl22anc 834 is also e11an 40903 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 40905 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 40906 Conjunction form of e01 40905. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   𝜓   ▶   𝜒   )    &   ((𝜑𝜒) → 𝜃)       (   𝜓   ▶   𝜃   )
 
Theoremee01an 40907 e01an 40906 without virtual deductions. sylancr 587 is also a form of e01an 40906 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 40908 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 40909 Conjunction form of e10 40908. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   𝜒    &   ((𝜓𝜒) → 𝜃)       (   𝜑   ▶   𝜃   )
 
Theoremee10an 40910 e10an 40909 without virtual deductions. sylancl 586 is also e10an 40909 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 40911 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 40912 Conjunction form of e02 40911. (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 40913 e02an 40912 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 40914 el021old 40915 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 40915 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 40916 syl2an 595 with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016.)
(𝜑𝜓)    &   ((𝜒𝜃) → 𝜏)    &   ((𝜓𝜏) → 𝜂)       ((𝜑𝜒𝜃) → 𝜂)
 
Theoremeel000cT 40917 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 40918 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 40919 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 40920 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 40921 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 40922 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 40923 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 40924 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 40925 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 40926 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 40927 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
(𝜑𝜓)    &   ((𝜑𝜒) → 𝜃)    &   ((𝜑𝜏) → 𝜂)    &   ((𝜓𝜃𝜂) → 𝜁)       ((𝜑𝜒𝜏) → 𝜁)
 
Theoremeel2131 40928 syl2an 595 with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜏)    &   ((𝜒𝜏) → 𝜂)       ((𝜑𝜓𝜃) → 𝜂)
 
Theoremeel3132 40929 syl2an 595 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
((𝜑𝜓) → 𝜒)    &   ((𝜃𝜓) → 𝜏)    &   ((𝜒𝜏) → 𝜂)       ((𝜑𝜃𝜓) → 𝜂)
 
Theoremeel0321old 40930 el0321old 40931 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 40931 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 40932 el2122old 40933 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 40933 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   )
 
Theoremeel0000 40934 Elimination rule similar to mp4an 689, except with a left-nested conjunction unification theorem. (Contributed by Alan Sare, 17-Oct-2017.)
𝜑    &   𝜓    &   𝜒    &   𝜃    &   ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       𝜏
 
Theoremeel00001 40935 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
𝜑    &   𝜓    &   𝜒    &   𝜃    &   (𝜏𝜂)    &   (((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜂) → 𝜁)       (𝜏𝜁)
 
Theoremeel00000 40936 Elimination rule similar eel0000 40934, except with five hpothesis steps. (Contributed by Alan Sare, 17-Oct-2017.)
𝜑    &   𝜓    &   𝜒    &   𝜃    &   𝜏    &   (((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)       𝜂
 
Theoremeel11111 40937 Five-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl113anc 1374 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 40938 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 40939 Conjunction form of e12 40938 (see syl6an 680). (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 40940 Virtual deduction form of syl2an 595. (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 40941 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 40942 Conjunction form of e20 40941. (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 40943 e20an 40942 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 40944 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 40945 Conjunction form of e21 40944. (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 40946 e21an 40945 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 40947 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 40948 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 40949 Conjunction form of e33 40948. (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 40950 e33an 40949 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 40951 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 40952 Biconditional form of e3 40951. syl8ib 257 is e3bi 40952 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 40953 Right biconditional form of e3 40951. (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 40954 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 40955 e03 40954 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 40956 Conjunction form of e03 40954. (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 40957 Conjunction form of ee03 40955. (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 40958 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 40959 e30 40958 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 40960 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 40961 Conjunction form of ee30 40959. (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 40962 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 40963 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 40964 e13an 40963 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 40965 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 40966 e31 40965 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 40967 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   )
 
Theoremee31an 40968 e31an 40967 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑𝜏)    &   ((𝜃𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒𝜂)))
 
Theoreme23 40969 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 → (𝜏𝜂))       (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )
 
Theoreme23an 40970 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   )    &   ((𝜒𝜏) → 𝜂)       (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )
 
Theoremee23an 40971 e23an 40970 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓 → (𝜃𝜏)))    &   ((𝜒𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜃𝜂)))
 
Theoreme32 40972 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 → (𝜏𝜂))       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
 
Theoremee32 40973 e32 40972 without virtual deductions. (Contributed by Alan Sare, 18-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓𝜏))    &   (𝜃 → (𝜏𝜂))       (𝜑 → (𝜓 → (𝜒𝜂)))
 
Theoreme32an 40974 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   )    &   ((𝜃𝜏) → 𝜂)       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
 
Theoremee32an 40975 e33an 40949 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓𝜏))    &   ((𝜃𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒𝜂)))
 
Theoreme123 40976 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 → (𝜂𝜁)))       (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜁   )
 
Theoremee123 40977 e123 40976 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜒 → (𝜏𝜂)))    &   (𝜓 → (𝜃 → (𝜂𝜁)))       (𝜑 → (𝜒 → (𝜏𝜁)))
 
Theoremel123 40978 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   )    &   ((𝜓𝜃𝜂) → 𝜁)       (   (   𝜑   ,   𝜒   ,   𝜏   )   ▶   𝜁   )
 
Theoreme233 40979 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 → (𝜂𝜁)))       (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜁   )
 
Theoreme323 40980 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 → (𝜂𝜁)))       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜁   )
 
Theoreme000 40981 A virtual deduction elimination rule. The non-virtual deduction form of e000 40981 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 → (𝜒𝜃)))       𝜃
 
Theoreme00 40982 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 → (𝜓𝜒))       𝜒
 
Theoreme00an 40983 Elimination rule identical to mp2an 688. The non-virtual deduction form is the virtual deduction form, which is mp2an 688. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   ((𝜑𝜓) → 𝜒)       𝜒
 
Theoremeel00cT 40984 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   ((𝜑𝜓) → 𝜒)       (⊤ → 𝜒)
 
TheoremeelTT 40985 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (⊤ → 𝜓)    &   ((𝜑𝜓) → 𝜒)       𝜒
 
Theoreme0a 40986 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    &   (𝜑𝜓)       𝜓
 
TheoremeelT 40987 An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (𝜑𝜓)       𝜓
 
Theoremeel0cT 40988 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       (⊤ → 𝜓)
 
TheoremeelT0 40989 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   𝜓    &   ((𝜑𝜓) → 𝜒)       𝜒
 
Theoreme0bi 40990 Elimination rule identical to mpbi 231. The non-virtual deduction form is the virtual deduction form, which is mpbi 231. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       𝜓
 
Theoreme0bir 40991 Elimination rule identical to mpbir 232. The non-virtual deduction form is the virtual deduction form, which is mpbir 232. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓𝜑)       𝜓
 
Theoremuun0.1 40992 Convention notation form of un0.1 40993. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (𝜓𝜒)    &   ((⊤ ∧ 𝜓) → 𝜃)       (𝜓𝜃)
 
Theoremun0.1 40993 is the constant true, a tautology (see df-tru 1531). 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   )
 
TheoremuunT1 40994 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 1531. (Revised by David A. Wheeler, 8-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ∧ 𝜑) → 𝜓)       (𝜑𝜓)
 
TheoremuunT1p1 40995 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)       (𝜑𝜓)
 
TheoremuunT21 40996 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.)
((⊤ ∧ (𝜑𝜓)) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremuun121 40997 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 ∧ (𝜑𝜓)) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremuun121p1 40998 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.)
(((𝜑𝜓) ∧ 𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremuun132 40999 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 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremuun132p1 41000 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.)
(((𝜓𝜒) ∧ 𝜑) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
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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 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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