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Theorem List for Metamath Proof Explorer - 41101-41200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiin3 41101 in3 41100 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theoremin3an 41102 The virtual deduction introduction rule converting the second conjunct of the third virtual hypothesis into the antecedent of the conclusion. exp4a 435 is the non-virtual deduction form of in3an 41102. (Contributed by Alan Sare, 25-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ,   (𝜒𝜃)   ▶   𝜏   )       (   𝜑   ,   𝜓   ,   𝜒   ▶   (𝜃𝜏)   )
 
Theoremint3 41103 The virtual deduction introduction rule of converting the end virtual hypothesis of 3 virtual hypotheses into an antecedent. Conventional form of int3 41103 is 3expia 1118. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (   𝜑   ,   𝜓   ,   𝜒   )   ▶   𝜃   )       (   (   𝜑   ,   𝜓   )   ▶   (𝜒𝜃)   )
 
Theoremidn2 41104 Virtual deduction identity rule which is idd 24 with virtual deduction symbols. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜓   )
 
Theoremiden2 41105 Virtual deduction identity rule. simpr 488 in conjunction form Virtual Deduction notation. (Contributed by Alan Sare, 5-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (   𝜑   ,   𝜓   )   ▶   𝜓   )
 
Theoremidn3 41106 Virtual deduction identity rule for three virtual hypotheses. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ,   𝜒   ▶   𝜒   )
 
Theoremgen11 41107* Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis. alrimiv 1929 is gen11 41107 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )       (   𝜑   ▶   𝑥𝜓   )
 
Theoremgen11nv 41108 Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis without distinct variables. alrimih 1825 is gen11nv 41108 without virtual deductions. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (   𝜑   ▶   𝜓   )       (   𝜑   ▶   𝑥𝜓   )
 
Theoremgen12 41109* Virtual deduction generalizing rule for two quantifying variables and one virtual hypothesis. gen12 41109 is alrimivv 1930 with virtual deductions. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )       (   𝜑   ▶   𝑥𝑦𝜓   )
 
Theoremgen21 41110* Virtual deduction generalizing rule for one quantifying variables and two virtual hypothesis. gen21 41110 is alrimdv 1931 with virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )       (   𝜑   ,   𝜓   ▶   𝑥𝜒   )
 
Theoremgen21nv 41111 Virtual deduction form of alrimdh 1865. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (   𝜑   ,   𝜓   ▶   𝜒   )       (   𝜑   ,   𝜓   ▶   𝑥𝜒   )
 
Theoremgen31 41112* Virtual deduction generalizing rule for one quantifying variable and three virtual hypothesis. gen31 41112 is ggen31 41036 with virtual deductions. (Contributed by Alan Sare, 22-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝑥𝜃   )
 
Theoremgen22 41113* Virtual deduction generalizing rule for two quantifying variables and two virtual hypothesis. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )       (   𝜑   ,   𝜓   ▶   𝑥𝑦𝜒   )
 
Theoremggen22 41114* gen22 41113 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝑦𝜒))
 
Theoremexinst 41115 Existential Instantiation. Virtual deduction form of exlimexi 41015. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (   𝑥𝜑   ,   𝜑   ▶   𝜓   )       (∃𝑥𝜑𝜓)
 
Theoremexinst01 41116 Existential Instantiation. Virtual Deduction rule corresponding to a special case of the Natural Deduction Sequent Calculus rule called Rule C in [Margaris] p. 79 and E in Table 1 on page 4 of the paper "Extracting information from intermediate T-systems" (2000) presented at IMLA99 by Mauro Ferrari, Camillo Fiorentini, and Pierangelo Miglioli. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (   𝜑   ,   𝜓   ▶   𝜒   )    &   (𝜑 → ∀𝑥𝜑)    &   (𝜒 → ∀𝑥𝜒)       (   𝜑   ▶   𝜒   )
 
Theoremexinst11 41117 Existential Instantiation. Virtual Deduction rule corresponding to a special case of the Natural Deduction Sequent Calculus rule called Rule C in [Margaris] p. 79 and E in Table 1 on page 4 of the paper "Extracting information from intermediate T-systems" (2000) presented at IMLA99 by Mauro Ferrari, Camillo Fiorentini, and Pierangelo Miglioli. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝑥𝜓   )    &   (   𝜑   ,   𝜓   ▶   𝜒   )    &   (𝜑 → ∀𝑥𝜑)    &   (𝜒 → ∀𝑥𝜒)       (   𝜑   ▶   𝜒   )
 
Theoreme1a 41118 A Virtual deduction elimination rule. syl 17 is e1a 41118 without virtual deductions. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (𝜓𝜒)       (   𝜑   ▶   𝜒   )
 
Theoremel1 41119 A Virtual deduction elimination rule. syl 17 is el1 41119 without virtual deductions. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (𝜓𝜒)       (   𝜑   ▶   𝜒   )
 
Theoreme1bi 41120 Biconditional form of e1a 41118. sylib 221 is e1bi 41120 without virtual deductions. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (𝜓𝜒)       (   𝜑   ▶   𝜒   )
 
Theoreme1bir 41121 Right biconditional form of e1a 41118. sylibr 237 is e1bir 41121 without virtual deductions. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (𝜒𝜓)       (   𝜑   ▶   𝜒   )
 
Theoreme2 41122 A virtual deduction elimination rule. syl6 35 is e2 41122 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (𝜒𝜃)       (   𝜑   ,   𝜓   ▶   𝜃   )
 
Theoreme2bi 41123 Biconditional form of e2 41122. syl6ib 254 is e2bi 41123 without virtual deductions. (Contributed by Alan Sare, 10-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (𝜒𝜃)       (   𝜑   ,   𝜓   ▶   𝜃   )
 
Theoreme2bir 41124 Right biconditional form of e2 41122. syl6ibr 255 is e2bir 41124 without virtual deductions. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (𝜃𝜒)       (   𝜑   ,   𝜓   ▶   𝜃   )
 
Theoremee223 41125 e223 41126 without virtual deductions. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜓 → (𝜏𝜂)))    &   (𝜒 → (𝜃 → (𝜂𝜁)))       (𝜑 → (𝜓 → (𝜏𝜁)))
 
Theoreme223 41126 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (   𝜑   ,   𝜓   ▶   𝜃   )    &   (   𝜑   ,   𝜓   ,   𝜏   ▶   𝜂   )    &   (𝜒 → (𝜃 → (𝜂𝜁)))       (   𝜑   ,   𝜓   ,   𝜏   ▶   𝜁   )
 
Theoreme222 41127 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (   𝜑   ,   𝜓   ▶   𝜃   )    &   (   𝜑   ,   𝜓   ▶   𝜏   )    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )
 
Theoreme220 41128 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 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )
 
Theoremee220 41129 e220 41128 without virtual deductions. (Contributed by Alan Sare, 12-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   𝜏    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))
 
Theoreme202 41130 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   )    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )
 
Theoremee202 41131 e202 41130 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   𝜃    &   (𝜑 → (𝜓𝜏))    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))
 
Theoreme022 41132 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   ▶   𝜏   )    &   (𝜑 → (𝜃 → (𝜏𝜂)))       (   𝜓   ,   𝜒   ▶   𝜂   )
 
Theoremee022 41133 e022 41132 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓 → (𝜒𝜃))    &   (𝜓 → (𝜒𝜏))    &   (𝜑 → (𝜃 → (𝜏𝜂)))       (𝜓 → (𝜒𝜂))
 
Theoreme002 41134 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   )    &   (𝜑 → (𝜓 → (𝜏𝜂)))       (   𝜒   ,   𝜃   ▶   𝜂   )
 
Theoremee002 41135 e002 41134 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   (𝜒 → (𝜃𝜏))    &   (𝜑 → (𝜓 → (𝜏𝜂)))       (𝜒 → (𝜃𝜂))
 
Theoreme020 41136 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    &   (𝜑 → (𝜃 → (𝜏𝜂)))       (   𝜓   ,   𝜒   ▶   𝜂   )
 
Theoremee020 41137 e020 41136 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓 → (𝜒𝜃))    &   𝜏    &   (𝜑 → (𝜃 → (𝜏𝜂)))       (𝜓 → (𝜒𝜂))
 
Theoreme200 41138 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    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )
 
Theoremee200 41139 e200 41138 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   𝜃    &   𝜏    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))
 
Theoreme221 41140 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   )    &   (   𝜑   ▶   𝜏   )    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )
 
Theoremee221 41141 e221 41140 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑𝜏)    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))
 
Theoreme212 41142 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   ,   𝜓   ▶   𝜏   )    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜓   ▶   𝜂   )
 
Theoremee212 41143 e212 41142 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)    &   (𝜑 → (𝜓𝜏))    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))
 
Theoreme122 41144 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   ,   𝜒   ▶   𝜏   )    &   (𝜓 → (𝜃 → (𝜏𝜂)))       (   𝜑   ,   𝜒   ▶   𝜂   )
 
Theoreme112 41145 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ▶   𝜒   )    &   (   𝜑   ,   𝜃   ▶   𝜏   )    &   (𝜓 → (𝜒 → (𝜏𝜂)))       (   𝜑   ,   𝜃   ▶   𝜂   )
 
Theoremee112 41146 e112 41145 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑 → (𝜃𝜏))    &   (𝜓 → (𝜒 → (𝜏𝜂)))       (𝜑 → (𝜃𝜂))
 
Theoreme121 41147 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 41148 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 41149 e211 41148 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 41150 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 41151 e210 41150 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 41152 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 41153 e201 41152 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 41154 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 41155 Virtual deduction rule e120 41154 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 41156 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 41157 e021 41156 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 41158 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 41159 e012 41158 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 41160 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 41161 e102 41160 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 41162 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 41163 Conjunction form of e22 41162. (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 41164 e22an 41163 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 41165 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 41166 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 41167 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 41168 e110 41167 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 41169 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 41170 e101 41169 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 41171 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 41172 e011 41171 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 41173 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 41174 e100 41173 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 41175 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 41176 e010 41175 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 41177 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 41178 e001 41177 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 41179 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 41180 Conjunction form of e11 41179. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ▶   𝜒   )    &   ((𝜓𝜒) → 𝜃)       (   𝜑   ▶   𝜃   )
 
Theoremee11an 41181 e11an 41180 without virtual deductions. syl22anc 837 is also e11an 41180 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 41182 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 41183 Conjunction form of e01 41182. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (   𝜓   ▶   𝜒   )    &   ((𝜑𝜒) → 𝜃)       (   𝜓   ▶   𝜃   )
 
Theoremee01an 41184 e01an 41183 without virtual deductions. sylancr 590 is also a form of e01an 41183 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 41185 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 41186 Conjunction form of e10 41185. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   𝜒    &   ((𝜓𝜒) → 𝜃)       (   𝜑   ▶   𝜃   )
 
Theoremee10an 41187 e10an 41186 without virtual deductions. sylancl 589 is also e10an 41186 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 41188 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 41189 Conjunction form of e02 41188. (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 41190 e02an 41189 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 41191 el021old 41192 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 41192 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 41193 syl2an 598 with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016.)
(𝜑𝜓)    &   ((𝜒𝜃) → 𝜏)    &   ((𝜓𝜏) → 𝜂)       ((𝜑𝜒𝜃) → 𝜂)
 
Theoremeel000cT 41194 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 41195 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 41196 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 41197 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 41198 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 41199 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 41200 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)
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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 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