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Theorem List for Metamath Proof Explorer - 27401-27500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdfvd2 27401 Definition of a 2-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( (. ph ,. ps  ->.  ch ).  <->  ( ph  ->  ( ps  ->  ch )
 ) )
 
Syntaxwvhc2 27402 Syntax for a 2-virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.)
 wff  (.
 ph ,. ps ).
 
Definitiondf-vhc2 27403 Definition of a 2-virtual hypotheses collection. (Contributed by Alan Sare, 23-Apr-2015.) (New usage is discouraged.)
 |-  ( (. ph ,. ps ).  <->  (
 ph  /\  ps )
 )
 
Theoremdfvd2an 27404 Definition of a 2-hypothesis virtual deduction in vd conjunction form. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( (. (. ph ,. ps ).  ->.  ch ).  <->  ( ( ph  /\ 
 ps )  ->  ch )
 )
 
Theoremdfvd2ani 27405 Inference form of dfvd2an 27404. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (. ph
 ,. ps ).  ->.  ch ).   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theoremdfvd2anir 27406 Right-to-left inference form of dfvd2an 27404. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |- 
 (. (. ph ,. ps ).  ->.  ch ).
 
Theoremdfvd2i 27407 Inference form of dfvd2 27401. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   =>    |-  ( ph  ->  ( ps  ->  ch ) )
 
Theoremdfvd2ir 27408 Right-to-left inference form of dfvd2 27401. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |- 
 (. ph ,. ps  ->.  ch ).
 
Syntaxwvd3 27409 Syntax for a 3-hypothesis virtual deduction. (New usage is discouraged.)
 wff  (.
 ph ,. ps ,. ch  ->.  th
 ).
 
Syntaxwvhc3 27410 Syntax for a 3-virtual hypotheses collection. (Contributed by Alan Sare, 13-Jun-2015.) (New usage is discouraged.)
 wff  (.
 ph ,. ps ,. ch ).
 
Definitiondf-vhc3 27411 Definition of a 3-virtual hypotheses collection. (Contributed by Alan Sare, 13-Jun-2015.) (New usage is discouraged.)
 |-  ( (. ph ,. ps ,. ch ).  <->  ( ph  /\  ps  /\ 
 ch ) )
 
Definitiondf-vd3 27412 Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (New usage is discouraged.)
 |-  ( (. ph ,. ps ,. ch  ->. 
 th ).  <->  ( ( ph  /\ 
 ps  /\  ch )  ->  th ) )
 
Theoremdfvd3 27413 Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( (. ph ,. ps ,. ch  ->. 
 th ).  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
 
Theoremdfvd3i 27414 Inference form of dfvd3 27413. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ch  ->.  th ).   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
 
Theoremdfvd3ir 27415 Right-to-left inference form of dfvd3 27413. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   =>    |- 
 (. ph ,. ps ,. ch  ->. 
 th ).
 
Theoremdfvd3an 27416 Definition of a 3-hypothesis virtual deduction in vd conjunction form. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( (. (. ph ,. ps ,. ch ).  ->.  th ).  <->  ( ( ph  /\ 
 ps  /\  ch )  ->  th ) )
 
Theoremdfvd3ani 27417 Inference form of dfvd3an 27416. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (. ph
 ,. ps ,. ch ).  ->.  th
 ).   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theoremdfvd3anir 27418 Right-to-left inference form of dfvd3an 27416. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (. (. ph
 ,. ps ,. ch ).  ->.  th
 ).
 
Theoremvd01 27419 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 27420 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 27421 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 27422 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 27423 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 27424 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 27425 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 27426 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 27427 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 27428 The virtual deduction introduction rule of converting the end virtual hypothesis of 2 virtual hypotheses into an antecedent. Conventional form of int2 27428 is ex 425. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (. ph
 ,. ps ).  ->.  ch ).   =>    |-  (. ph  ->.  ( ps  ->  ch ) ).
 
Theoremiin2 27429 in2 27427 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 27430 The virtual deduction introduction rule converting the second conjunct of the second virtual hypothesis into the antecedent of the conclusion. exp3a 427 is the non-virtual deduction form of in2an 27430. (Contributed by Alan Sare, 30-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ( ps  /\  ch ) 
 ->.  th ).   =>    |- 
 (. ph ,. ps  ->.  ( ch  ->  th ) ).
 
Theoremin3 27431 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 27432 in3 27431 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 27433 The virtual deduction introduction rule converting the second conjunct of the third virtual hypothesis into the antecedent of the conclusion. exp4a 592 is the non-virtual deduction form of in3an 27433. (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 27434 The virtual deduction introduction rule of converting the end virtual hypothesis of 3 virtual hypotheses into an antecedent. Conventional form of int3 27434 is 3expia 1158. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (. ph
 ,. ps ,. ch ).  ->.  th
 ).   =>    |- 
 (. (. ph ,. ps ).  ->.  ( ch  ->  th ) ).
 
Theoremidn2 27435 Virtual deduction identity rule which is idd 23 with virtual deduction symbols. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ps ).
 
Theoremiden2 27436 Virtual deduction identity rule. simpr 449 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 27437 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 27438* Virtual deduction generalizing rule for 1 quantifying variable and 1 virtual hypothesis. alrimiv 2013 is gen11 27438 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 27439 Virtual deduction generalizing rule for 1 quantifying variable and 1 virtual hypothesis without distinct variables. alrimih 1553 is gen11nv 27439 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 27440* Virtual deduction generalizing rule for 2 quantifying variables and 1 virtual hypothesis. gen12 27440 is alrimivv 2014 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 27441* Virtual deduction generalizing rule for 1 quantifying variables and 2 virtual hypothesis. gen21 27441 is alrimdv 2015 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 27442 Virtual deduction form of alrimdh 1585. (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 27443* Virtual deduction generalizing rule for 1 quantifying variable and 3 virtual hypothesis. gen31 27443 is ggen31 27363 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 27444* 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 27445* gen22 27444 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 27446 Existential Instantiation. Virtual deduction form of exlimexi 27340. (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 27447 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 27448 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_ 27449 A Virtual deduction elimination rule. syl 17 is e1_ 27449 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 27450 A Virtual deduction elimination rule. syl 17 is el1 27450 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 27451 Biconditional form of e1_ 27449. sylib 190 is e1bi 27451 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 27452 Right biconditional form of e1_ 27449. sylibr 205 is e1bir 27452 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 27453 A virtual deduction elimination rule. syl6 31 is e2 27453 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 27454 Bi-conditional form of e2 27453. syl6ib 219 is e2bi 27454 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 27455 Right bi-conditional form of e2 27453. syl6ibr 220 is e2bir 27455 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 27456 e223 27457 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 27457 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 27458 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 27459 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 27460 e220 27459 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 27461 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 27462 e202 27461 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 27463 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 27464 e022 27463 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 27465 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 27466 e002 27465 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 27467 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 27468 e020 27467 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 27469 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 27470 e200 27469 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 27471 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 27472 e221 27471 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 27473 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 27474 e212 27473 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 27475 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 27476 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 27477 e112 27476 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 27478 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 27479 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 27480 e211 27479 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 27481 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 27482 e210 27481 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 27483 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 27484 e201 27483 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 27485 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 27486 Virtual deduction rule e120 27485 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 27487 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 27488 e021 27487 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 27489 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 27490 e012 27489 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 27491 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 27492 e102 27491 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 27493 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 27494 Conjunction form of e22 27493. (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 27495 e22an 27494 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 27496 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 27497 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 27498 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 27499 e110 27498 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 27500 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 ).
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