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Theorem List for Metamath Proof Explorer - 28401-28500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxwvhc7 28401 Syntax for a 7-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.)
 wff  (.
 ph ,. ps ,. ch ,. th ,. ta ,. et ,. ze ).
 
Syntaxwvhc8 28402 Syntax for an 8-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.)
 wff  (.
 ph ,. ps ,. ch ,. th ,. ta ,. et ,. ze ,. si ).
 
Syntaxwvhc9 28403 Syntax for a 9-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.)
 wff  (.
 ph ,. ps ,. ch ,. th ,. ta ,. et ,. ze ,. si ,. rh ).
 
Syntaxwvhc10 28404 Syntax for a 10-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.)
 wff  (.
 ph ,. ps ,. ch ,. th ,. ta ,. et ,. ze ,. si ,. rh ,. mu ).
 
Syntaxwvhc11 28405 Syntax for an 11-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.)
 wff  (.
 ph ,. ps ,. ch ,. th ,. ta ,. et ,. ze ,. si ,. rh ,. mu ,. la
 ).
 
Syntaxwvhc12 28406 Syntax for an 12-element virtual hypotheses collection. (Contributed by Alan Sare, 17-Oct-2017.) (New usage is discouraged.)
 wff  (.
 ph ,. ps ,. ch ,. th ,. ta ,. et ,. ze ,. si ,. rh ,. mu ,. la
 ,. ka ).
 
Theoremvd01 28407 A virtual hypothesis virtually infers a theorem. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   =>    |- 
 (. ps  ->.  ph ).
 
Theoremvd02 28408 2 virtual hypotheses virtually infer a theorem. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   =>    |- 
 (. ps ,. ch  ->.  ph ).
 
Theoremvd03 28409 A theorem is virtually inferred by the 3 virtual hypotheses. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   =>    |- 
 (. ps ,. ch ,. th  ->. 
 ph ).
 
Theoremvd12 28410 A virtual deduction with 1 virtual hypothesis virtually inferring a virtual conclusion infers that the same conclusion is virtually inferred by the same virtual hypothesis and an additional hypothesis. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   =>    |- 
 (. ph ,. ch  ->.  ps ).
 
Theoremvd13 28411 A virtual deduction with 1 virtual hypothesis virtually inferring a virtual conclusion infers that the same conclusion is virtually inferred by the same virtual hypothesis and a two additional hypotheses. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   =>    |- 
 (. ph ,. ch ,. th  ->.  ps ).
 
Theoremvd23 28412 A virtual deduction with 2 virtual hypotheses virtually inferring a virtual conclusion infers that the same conclusion is virtually inferred by the same 2 virtual hypotheses and a third hypothesis. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   =>    |- 
 (. ph ,. ps ,. th  ->.  ch ).
 
Theoremdfvd2imp 28413 The virtual deduction form of a 2-antecedent nested implication implies the 2-antecedent nested implication. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( (. ph ,. ps  ->.  ch ).  ->  (
 ph  ->  ( ps  ->  ch ) ) )
 
Theoremdfvd2impr 28414 A 2-antecedent nested implication implies its virtual deduction form. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  (. ph ,. ps  ->.  ch ). )
 
Theoremin2 28415 The virtual deduction introduction rule of converting the end virtual hypothesis of 2 virtual hypotheses into an antecedent. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   =>    |- 
 (. ph  ->.  ( ps  ->  ch ) ).
 
Theoremint2 28416 The virtual deduction introduction rule of converting the end virtual hypothesis of 2 virtual hypotheses into an antecedent. Conventional form of int2 28416 is ex 424. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (. ph
 ,. ps ).  ->.  ch ).   =>    |-  (. ph  ->.  ( ps  ->  ch ) ).
 
Theoremiin2 28417 in2 28415 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 28418 The virtual deduction introduction rule converting the second conjunct of the second virtual hypothesis into the antecedent of the conclusion. exp3a 426 is the non-virtual deduction form of in2an 28418. (Contributed by Alan Sare, 30-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ( ps  /\  ch ) 
 ->.  th ).   =>    |- 
 (. ph ,. ps  ->.  ( ch  ->  th ) ).
 
Theoremin3 28419 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 28420 in3 28419 without virtual deduction connectives. Special theorem needed for Alan Sare's 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 28421 The virtual deduction introduction rule converting the second conjunct of the third virtual hypothesis into the antecedent of the conclusion. exp4a 590 is the non-virtual deduction form of in3an 28421. (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 28422 The virtual deduction introduction rule of converting the end virtual hypothesis of 3 virtual hypotheses into an antecedent. Conventional form of int3 28422 is 3expia 1155. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (. ph
 ,. ps ,. ch ).  ->.  th
 ).   =>    |- 
 (. (. ph ,. ps ).  ->.  ( ch  ->  th ) ).
 
Theoremidn2 28423 Virtual deduction identity rule which is idd 22 with virtual deduction symbols. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ps ).
 
Theoremiden2 28424 Virtual deduction identity rule. simpr 448 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 28425 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 28426* Virtual deduction generalizing rule for 1 quantifying variable and 1 virtual hypothesis. alrimiv 1638 is gen11 28426 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 28427 Virtual deduction generalizing rule for 1 quantifying variable and 1 virtual hypothesis without distinct variables. alrimih 1571 is gen11nv 28427 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 28428* Virtual deduction generalizing rule for 2 quantifying variables and 1 virtual hypothesis. gen12 28428 is alrimivv 1639 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 28429* Virtual deduction generalizing rule for 1 quantifying variables and 2 virtual hypothesis. gen21 28429 is alrimdv 1640 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 28430 Virtual deduction form of alrimdh 1594. (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 28431* Virtual deduction generalizing rule for 1 quantifying variable and 3 virtual hypothesis. gen31 28431 is ggen31 28342 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 28432* 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 28433* gen22 28432 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 28434 Existential Instantiation. Virtual deduction form of exlimexi 28319. (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 28435 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 28436 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
 ).
 
Theoreme1_ 28437 A Virtual deduction elimination rule. syl 16 is e1_ 28437 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 28438 A Virtual deduction elimination rule. syl 16 is el1 28438 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 28439 Biconditional form of e1_ 28437. sylib 189 is e1bi 28439 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 28440 Right biconditional form of e1_ 28437. sylibr 204 is e1bir 28440 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 28441 A virtual deduction elimination rule. syl6 31 is e2 28441 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 28442 Bi-conditional form of e2 28441. syl6ib 218 is e2bi 28442 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 28443 Right bi-conditional form of e2 28441. syl6ibr 219 is e2bir 28443 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 28444 e223 28445 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 28445 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 28446 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 28447 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 28448 e220 28447 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 28449 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 28450 e202 28449 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 28451 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 28452 e022 28451 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 28453 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 28454 e002 28453 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 28455 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 28456 e020 28455 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 28457 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 28458 e200 28457 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 28459 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 28460 e221 28459 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 28461 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 28462 e212 28461 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 28463 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 28464 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 28465 e112 28464 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 28466 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 28467 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 28468 e211 28467 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 28469 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 28470 e210 28469 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 28471 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 28472 e201 28471 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 28473 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 28474 Virtual deduction rule e120 28473 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 28475 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 28476 e021 28475 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 28477 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 28478 e012 28477 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 28479 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 28480 e102 28479 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 28481 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 28482 Conjunction form of e22 28481. (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 28483 e22an 28482 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 28484 A virtual deduction elimination rule. (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 28485 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 28486 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 28487 e110 28486 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 28488 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 28489 e101 28488 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 28490 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 28491 e011 28490 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 28492 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 28493 e100 28492 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 28494 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 28495 e010 28494 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 28496 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 28497 e001 28496 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 28498 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 28499 Conjunction form of e11 28498. (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 28500 e11an 28499 without virtual deductions. syl22anc 1185 is also e11an 28499 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 )
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