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Theorem List for Metamath Proof Explorer - 33101-33200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiin2 33101 in2 33099 without virtual deductions. (Contributed by Alan Sare, 20-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  ch ) )
 
Theoremin2an 33102 The virtual deduction introduction rule converting the second conjunct of the second virtual hypothesis into the antecedent of the conclusion. expd 436 is the non-virtual deduction form of in2an 33102. (Contributed by Alan Sare, 30-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ( ps  /\  ch ) 
 ->.  th ).   =>    |- 
 (. ph ,. ps  ->.  ( ch  ->  th ) ).
 
Theoremin3 33103 The virtual deduction introduction rule of converting the end virtual hypothesis of 3 virtual hypotheses into an antecedent. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ch  ->.  th ).   =>    |-  (. ph ,. ps  ->.  ( ch  ->  th ) ).
 
Theoremiin3 33104 in3 33103 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.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
 
Theoremin3an 33105 The virtual deduction introduction rule converting the second conjunct of the third virtual hypothesis into the antecedent of the conclusion. exp4a 606 is the non-virtual deduction form of in3an 33105. (Contributed by Alan Sare, 25-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ( ch 
 /\  th )  ->.  ta ).   =>    |-  (. ph ,. ps ,. ch  ->.  ( th  ->  ta ) ).
 
Theoremint3 33106 The virtual deduction introduction rule of converting the end virtual hypothesis of 3 virtual hypotheses into an antecedent. Conventional form of int3 33106 is 3expia 1197. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (. ph
 ,. ps ,. ch ).  ->.  th
 ).   =>    |- 
 (. (. ph ,. ps ).  ->.  ( ch  ->  th ) ).
 
Theoremidn2 33107 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.)
 |-  (. ph ,. ps  ->.  ps ).
 
Theoremiden2 33108 Virtual deduction identity rule. simpr 461 in conjunction form Virtual Deduction notation. (Contributed by Alan Sare, 5-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (. ph
 ,. ps ).  ->.  ps ).
 
Theoremidn3 33109 Virtual deduction identity rule for 3 virtual hypotheses. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ch  ->.  ch ).
 
Theoremgen11 33110* Virtual deduction generalizing rule for 1 quantifying variable and 1 virtual hypothesis. alrimiv 1704 is gen11 33110 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   =>    |- 
 (. ph  ->.  A. x ps ).
 
Theoremgen11nv 33111 Virtual deduction generalizing rule for 1 quantifying variable and 1 virtual hypothesis without distinct variables. alrimih 1627 is gen11nv 33111 without virtual deductions. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  (. ph  ->.  ps
 ).   =>    |- 
 (. ph  ->.  A. x ps ).
 
Theoremgen12 33112* Virtual deduction generalizing rule for 2 quantifying variables and 1 virtual hypothesis. gen12 33112 is alrimivv 1705 with virtual deductions. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   =>    |- 
 (. ph  ->.  A. x A. y ps ).
 
Theoremgen21 33113* Virtual deduction generalizing rule for 1 quantifying variables and 2 virtual hypothesis. gen21 33113 is alrimdv 1706 with virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   =>    |- 
 (. ph ,. ps  ->.  A. x ch ).
 
Theoremgen21nv 33114 Virtual deduction form of alrimdh 1657. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   &    |-  (. ph ,. ps  ->.  ch ).   =>    |- 
 (. ph ,. ps  ->.  A. x ch ).
 
Theoremgen31 33115* Virtual deduction generalizing rule for 1 quantifying variable and 3 virtual hypothesis. gen31 33115 is ggen31 33025 with virtual deductions. (Contributed by Alan Sare, 22-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ch  ->.  th ).   =>    |-  (. ph ,. ps ,. ch  ->.  A. x th ).
 
Theoremgen22 33116* Virtual deduction generalizing rule for 2 quantifying variables and 2 virtual hypothesis. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   =>    |- 
 (. ph ,. ps  ->.  A. x A. y ch ).
 
Theoremggen22 33117* gen22 33116 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x A. y ch ) )
 
Theoremexinst 33118 Existential Instantiation. Virtual deduction form of exlimexi 33002. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   &    |-  (. E. x ph ,. ph  ->.  ps ).   =>    |-  ( E. x ph 
 ->  ps )
 
Theoremexinst01 33119 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  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.)
 |-  E. x ps   &    |- 
 (. ph ,. ps  ->.  ch ).   &    |-  ( ph  ->  A. x ph )   &    |-  ( ch  ->  A. x ch )   =>    |-  (. ph  ->.  ch
 ).
 
Theoremexinst11 33120 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  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.)
 |-  (. ph  ->.  E. x ps ).   &    |-  (. ph ,. ps  ->.  ch ).   &    |-  ( ph  ->  A. x ph )   &    |-  ( ch  ->  A. x ch )   =>    |-  (. ph  ->.  ch
 ).
 
Theoreme1a 33121 A Virtual deduction elimination rule. syl 16 is e1a 33121 without virtual deductions. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |-  ( ps  ->  ch )   =>    |-  (. ph  ->.  ch
 ).
 
Theoremel1 33122 A Virtual deduction elimination rule. syl 16 is el1 33122 without virtual deductions. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |-  ( ps  ->  ch )   =>    |-  (. ph  ->.  ch
 ).
 
Theoreme1bi 33123 Biconditional form of e1a 33121. sylib 196 is e1bi 33123 without virtual deductions. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |-  ( ps  <->  ch )   =>    |- 
 (. ph  ->.  ch ).
 
Theoreme1bir 33124 Right biconditional form of e1a 33121. sylibr 212 is e1bir 33124 without virtual deductions. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |-  ( ch  <->  ps )   =>    |- 
 (. ph  ->.  ch ).
 
Theoreme2 33125 A virtual deduction elimination rule. syl6 33 is e2 33125 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  ( ch  ->  th )   =>    |- 
 (. ph ,. ps  ->.  th ).
 
Theoreme2bi 33126 Biconditional form of e2 33125. syl6ib 226 is e2bi 33126 without virtual deductions. (Contributed by Alan Sare, 10-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  ( ch  <->  th )   =>    |- 
 (. ph ,. ps  ->.  th ).
 
Theoreme2bir 33127 Right biconditional form of e2 33125. syl6ibr 227 is e2bir 33127 without virtual deductions. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  ( th  <->  ch )   =>    |- 
 (. ph ,. ps  ->.  th ).
 
Theoremee223 33128 e223 33129 without virtual deductions. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th )
 )   &    |-  ( ph  ->  ( ps  ->  ( ta  ->  et ) ) )   &    |-  ( ch  ->  ( th  ->  ( et  ->  ze )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ta  ->  ze ) ) )
 
Theoreme223 33129 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph ,. ps  ->. 
 th ).   &    |-  (. ph ,. ps ,. ta  ->.  et ).   &    |-  ( ch  ->  ( th  ->  ( et  ->  ze )
 ) )   =>    |- 
 (. ph ,. ps ,. ta  ->.  ze ).
 
Theoreme222 33130 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph ,. ps  ->. 
 th ).   &    |-  (. ph ,. ps  ->.  ta ).   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. ps  ->.  et ).
 
Theoreme220 33131 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph ,. ps  ->. 
 th ).   &    |-  ta   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. ps  ->.  et ).
 
Theoremee220 33132 e220 33131 without virtual deductions. (Contributed by Alan Sare, 12-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th )
 )   &    |- 
 ta   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et )
 )
 
Theoreme202 33133 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  th   &    |-  (. ph ,. ps  ->.  ta ).   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. ps  ->.  et ).
 
Theoremee202 33134 e202 33133 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  th   &    |-  ( ph  ->  ( ps  ->  ta )
 )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et )
 )
 
Theoreme022 33135 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps ,. ch  ->. 
 th ).   &    |-  (. ps ,. ch  ->.  ta ).   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ps ,. ch  ->.  et ).
 
Theoremee022 33136 e022 33135 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ( ch  ->  th )
 )   &    |-  ( ps  ->  ( ch  ->  ta ) )   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ps  ->  ( ch  ->  et ) )
 
Theoreme002 33137 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  (. ch ,. th  ->.  ta ).   &    |-  ( ph  ->  ( ps  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ch ,. th  ->.  et ).
 
Theoremee002 33138 e002 33137 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  ( ch  ->  ( th  ->  ta )
 )   &    |-  ( ph  ->  ( ps  ->  ( ta  ->  et ) ) )   =>    |-  ( ch  ->  ( th  ->  et )
 )
 
Theoreme020 33139 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps ,. ch  ->. 
 th ).   &    |-  ta   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ps ,. ch  ->.  et ).
 
Theoremee020 33140 e020 33139 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ( ch  ->  th )
 )   &    |- 
 ta   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ps  ->  ( ch  ->  et )
 )
 
Theoreme200 33141 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  th   &    |-  ta   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. ps  ->.  et ).
 
Theoremee200 33142 e200 33141 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  th   &    |-  ta   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoreme221 33143 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph ,. ps  ->. 
 th ).   &    |-  (. ph  ->.  ta ).   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ph ,. ps  ->.  et ).
 
Theoremee221 33144 e221 33143 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th )
 )   &    |-  ( ph  ->  ta )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoreme212 33145 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph  ->.  th ).   &    |-  (. ph ,. ps  ->.  ta ).   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. ps  ->.  et ).
 
Theoremee212 33146 e212 33145 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ( ps  ->  ta )
 )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et )
 )
 
Theoreme122 33147 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph ,. ch  ->.  th ).   &    |-  (. ph ,. ch  ->.  ta ).   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. ch  ->.  et ).
 
Theoreme112 33148 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph  ->.  ch ).   &    |-  (. ph ,. th  ->.  ta ).   &    |-  ( ps  ->  ( ch  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. th  ->.  et ).
 
Theoremee112 33149 e112 33148 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( th  ->  ta )
 )   &    |-  ( ps  ->  ( ch  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( th  ->  et )
 )
 
Theoreme121 33150 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph ,. ch  ->.  th ).   &    |-  (. ph  ->.  ta
 ).   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (. ph ,. ch  ->.  et ).
 
Theoreme211 33151 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph  ->.  th ).   &    |-  (. ph  ->.  ta
 ).   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (. ph ,. ps  ->.  et ).
 
Theoremee211 33152 e211 33151 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoreme210 33153 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph  ->.  th ).   &    |-  ta   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ph ,. ps  ->.  et ).
 
Theoremee210 33154 e210 33153 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  th )   &    |-  ta   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoreme201 33155 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  th   &    |-  (. ph  ->.  ta ).   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ph ,. ps  ->.  et ).
 
Theoremee201 33156 e201 33155 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  th   &    |-  ( ph  ->  ta )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoreme120 33157 A virtual deduction elimination rule. (Contributed by Alan Sare, 10-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph ,. ch  ->.  th ).   &    |-  ta   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ph ,. ch  ->.  et ).
 
Theoremee120 33158 Virtual deduction rule e120 33157 without virtual deduction symbols. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   &    |- 
 ta   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ch  ->  et )
 )
 
Theoreme021 33159 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps ,. ch  ->. 
 th ).   &    |-  (. ps  ->.  ta ).   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ps ,. ch  ->.  et ).
 
Theoremee021 33160 e021 33159 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ( ch  ->  th )
 )   &    |-  ( ps  ->  ta )   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ps  ->  ( ch  ->  et ) )
 
Theoreme012 33161 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps  ->.  ch ).   &    |-  (. ps ,. th  ->.  ta ).   &    |-  ( ph  ->  ( ch  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ps ,. th  ->.  et ).
 
Theoremee012 33162 e012 33161 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  ( ps  ->  ( th  ->  ta )
 )   &    |-  ( ph  ->  ( ch  ->  ( ta  ->  et ) ) )   =>    |-  ( ps  ->  ( th  ->  et )
 )
 
Theoreme102 33163 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 ch   &    |- 
 (. ph ,. th  ->.  ta ).   &    |-  ( ps  ->  ( ch  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ph ,. th  ->.  et ).
 
Theoremee102 33164 e102 33163 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  ( ph  ->  ( th  ->  ta )
 )   &    |-  ( ps  ->  ( ch  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( th  ->  et )
 )
 
Theoreme22 33165 A virtual deduction elimination rule. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph ,. ps  ->. 
 th ).   &    |-  ( ch  ->  ( th  ->  ta )
 )   =>    |- 
 (. ph ,. ps  ->.  ta ).
 
Theoreme22an 33166 Conjunction form of e22 33165. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph ,. ps  ->. 
 th ).   &    |-  ( ( ch 
 /\  th )  ->  ta )   =>    |-  (. ph ,. ps  ->.  ta ).
 
Theoremee22an 33167 e22an 33166 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th )
 )   &    |-  ( ( ch  /\  th )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ta ) )
 
Theoreme111 33168 A virtual deduction elimination rule (see syl3c 61). (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph  ->.  ch ).   &    |-  (. ph  ->.  th ).   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) )   =>    |- 
 (. ph  ->.  ta ).
 
Theoreme1111 33169 A virtual deduction elimination rule. (Contributed by Alan Sare, 6-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph  ->.  ch ).   &    |-  (. ph  ->.  th ).   &    |-  (. ph  ->.  ta
 ).   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) )   =>    |-  (. ph  ->.  et ).
 
Theoreme110 33170 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph  ->.  ch ).   &    |-  th   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  (.
 ph 
 ->.  ta ).
 
Theoremee110 33171 e110 33170 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  th   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  ( ph  ->  ta )
 
Theoreme101 33172 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 ch   &    |- 
 (. ph  ->.  th ).   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  (.
 ph 
 ->.  ta ).
 
Theoremee101 33173 e101 33172 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  ( ph  ->  th )   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  ( ph  ->  ta )
 
Theoreme011 33174 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps  ->.  ch ).   &    |-  (. ps  ->.  th
 ).   &    |-  ( ph  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  (. ps  ->.  ta ).
 
Theoremee011 33175 e011 33174 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  ( ps  ->  th )   &    |-  ( ph  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  ( ps  ->  ta )
 
Theoreme100 33176 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 ch   &    |- 
 th   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  (. ph  ->.  ta ).
 
Theoremee100 33177 e100 33176 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  th   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  ( ph  ->  ta )
 
Theoreme010 33178 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps  ->.  ch ).   &    |-  th   &    |-  ( ph  ->  ( ch  ->  ( th  ->  ta )
 ) )   =>    |- 
 (. ps  ->.  ta ).
 
Theoremee010 33179 e010 33178 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  th   &    |-  ( ph  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  ( ps  ->  ta )
 
Theoreme001 33180 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  (. ch  ->.  th ).   &    |-  ( ph  ->  ( ps  ->  ( th  ->  ta )
 ) )   =>    |- 
 (. ch  ->.  ta ).
 
Theoremee001 33181 e001 33180 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  ( ch  ->  th )   &    |-  ( ph  ->  ( ps  ->  ( th  ->  ta ) ) )   =>    |-  ( ch  ->  ta )
 
Theoreme11 33182 A virtual deduction elimination rule. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph  ->.  ch ).   &    |-  ( ps  ->  ( ch  ->  th )
 )   =>    |- 
 (. ph  ->.  th ).
 
Theoreme11an 33183 Conjunction form of e11 33182. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph  ->.  ch ).   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (. ph  ->.  th
 ).
 
Theoremee11an 33184 e11an 33183 without virtual deductions. syl22anc 1228 is also e11an 33183 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.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoreme01 33185 A virtual deduction elimination rule. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps  ->.  ch ).   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |- 
 (. ps  ->.  th ).
 
Theoreme01an 33186 Conjunction form of e01 33185. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps  ->.  ch ).   &    |-  (
 ( ph  /\  ch )  ->  th )   =>    |- 
 (. ps  ->.  th ).
 
Theoremee01an 33187 e01an 33186 without virtual deductions. sylancr 663 is also a form of e01an 33186 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.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  ( ( ph  /\ 
 ch )  ->  th )   =>    |-  ( ps  ->  th )
 
Theoreme10 33188 A virtual deduction elimination rule (see mpisyl 18). (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 ch   &    |-  ( ps  ->  ( ch  ->  th ) )   =>    |-  (. ph  ->.  th ).
 
Theoreme10an 33189 Conjunction form of e10 33188. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 ch   &    |-  ( ( ps  /\  ch )  ->  th )   =>    |-  (. ph  ->.  th
 ).
 
Theoremee10an 33190 e10an 33189 without virtual deductions. sylancl 662 is also e10an 33189 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.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoreme02 33191 A virtual deduction elimination rule. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps ,. ch  ->. 
 th ).   &    |-  ( ph  ->  ( th  ->  ta )
 )   =>    |- 
 (. ps ,. ch  ->.  ta ).
 
Theoreme02an 33192 Conjunction form of e02 33191. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps ,. ch  ->. 
 th ).   &    |-  ( ( ph  /\ 
 th )  ->  ta )   =>    |-  (. ps ,. ch  ->.  ta ).
 
Theoremee02an 33193 e02an 33192 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ( ch  ->  th )
 )   &    |-  ( ( ph  /\  th )  ->  ta )   =>    |-  ( ps  ->  ( ch  ->  ta ) )
 
Theoremeel021old 33194 el021old 33195 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ( ps 
 /\  ch )  ->  th )   &    |-  (
 ( ph  /\  th )  ->  ta )   =>    |-  ( ( ps  /\  ch )  ->  ta )
 
Theoremel021old 33195 A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. (. ps ,. ch ).  ->.  th ).   &    |-  (
 ( ph  /\  th )  ->  ta )   =>    |- 
 (. (. ps ,. ch ).  ->.  ta ).
 
Theoremeel132 33196 syl2an 477 with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016.)
 |-  ( ph  ->  ps )   &    |-  ( ( ch 
 /\  th )  ->  ta )   &    |-  (
 ( ps  /\  ta )  ->  et )   =>    |-  ( ( ph  /\ 
 ch  /\  th )  ->  et )
 
Theoremeel2221 33197 Deduction related to syl3an 1269 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ( th  /\  ph )  ->  ta )   &    |-  (
 ( ps  /\  ch  /\ 
 ta )  ->  et )   =>    |-  (
 ( th  /\  ph )  ->  et )
 
Theoremeel112 33198 syl3an 1269 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( th  ->  ta )   &    |-  ( ( ps 
 /\  ch  /\  ta )  ->  et )   =>    |-  ( ( ph  /\  th )  ->  et )
 
Theoremeel000cT 33199 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  ch   &    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )   =>    |-  ( T.  ->  th )
 
Theoremeel0TT 33200 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( T.  ->  ps )   &    |-  ( T.  ->  ch )   &    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )   =>    |- 
 th
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