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Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremee101 27501 e101 27500 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 27502 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 27503 e011 27502 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 27504 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 27505 e100 27504 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 27506 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 27507 e010 27506 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 27508 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 27509 e001 27508 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 27510 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 27511 Conjunction form of e11 27510. (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 27512 e11an 27511 without virtual deductions. syl22anc 1188 is also e11an 27511 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 27513 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 ).
 
Theoremee01OLD 27514 e01 27513 without virtual deduction symbols. (Moved to mpsyl 61 in main set.mm and may be deleted by mathbox owner, AS. --NM 22-Mar-2013.) (Contributed by Alan Sare, 20-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  ( ph  ->  ( ch  ->  th )
 )   =>    |-  ( ps  ->  th )
 
Theoreme01an 27515 Conjunction form of e01 27513. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps  ->.  ch ).   &    |-  (
 ( ph  /\  ch )  ->  th )   =>    |- 
 (. ps  ->.  th ).
 
Theoremee01an 27516 e01an 27515 without virtual deductions. sylancr 647 is also a form of e01an 27515 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 27517 A virtual deduction elimination rule. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 ch   &    |-  ( ps  ->  ( ch  ->  th ) )   =>    |-  (. ph  ->.  th ).
 
Theoreme10an 27518 Conjunction form of e10 27517. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 ch   &    |-  ( ( ps  /\  ch )  ->  th )   =>    |-  (. ph  ->.  th
 ).
 
Theoremee10an 27519 e10an 27518 without virtual deductions. sylancl 646 is also e10an 27518 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 27520 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 27521 Conjunction form of e02 27520. (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 27522 e02an 27521 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 27523 el021old 27524 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 27524 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 27525 syl2an 465 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 27526 Deduction related to to syl3an 1229 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 27527 syl3an 1229 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 27528 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 27529 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
 
TheoremeelT00 27530 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  T.  ->  ph )   &    |-  ps   &    |-  ch   &    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )   =>    |- 
 th
 
TheoremeelTTT 27531 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  T.  ->  ph )   &    |-  (  T.  ->  ps )   &    |-  (  T.  ->  ch )   &    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )   =>    |- 
 th
 
Theoremeel011 27532 mp3an 1282 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  ( ps  ->  th )   &    |-  ( ( ph  /\ 
 ch  /\  th )  ->  ta )   =>    |-  ( ps  ->  ta )
 
TheoremeelT11 27533 A elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  T.  ->  ph )   &    |-  ( ps  ->  ch )   &    |-  ( ps  ->  th )   &    |-  ( ( ph  /\ 
 ch  /\  th )  ->  ta )   =>    |-  ( ps  ->  ta )
 
Theoremeel012 27534 mp3an 1282 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  ( th  ->  ta )   &    |-  ( ( ph  /\ 
 ch  /\  ta )  ->  et )   =>    |-  ( ( ps  /\  th )  ->  et )
 
TheoremeelT1 27535 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Alan Sare, 23-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  T.  ->  ph )   &    |-  ( ps  ->  ch )   &    |-  ( ( ph  /\ 
 ch )  ->  th )   =>    |-  ( ps  ->  th )
 
TheoremeelT12 27536 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  T.  ->  ph )   &    |-  ( ps  ->  ch )   &    |-  ( th  ->  ta )   &    |-  ( ( ph  /\ 
 ch  /\  ta )  ->  et )   =>    |-  ( ( ps  /\  th )  ->  et )
 
Theoremeel001 27537 mp3an 1282 with antecedents in standard conjunction form and with one hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.)
 |-  ph   &    |-  ps   &    |-  ( ch  ->  th )   &    |-  ( ( ph  /\ 
 ps  /\  th )  ->  ta )   =>    |-  ( ch  ->  ta )
 
TheoremeelTT1 27538 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  T.  ->  ph )   &    |-  (  T.  ->  ps )   &    |-  ( ch  ->  th )   &    |-  ( ( ph  /\ 
 ps  /\  th )  ->  ta )   =>    |-  ( ch  ->  ta )
 
TheoremeelT01 27539 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  T.  ->  ph )   &    |-  ps   &    |-  ( ch  ->  th )   &    |-  ( ( ph  /\ 
 ps  /\  th )  ->  ta )   =>    |-  ( ch  ->  ta )
 
Theoremeel0T1 27540 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (  T.  ->  ps )   &    |-  ( ch  ->  th )   &    |-  ( ( ph  /\ 
 ps  /\  th )  ->  ta )   =>    |-  ( ch  ->  ta )
 
Theoremeel121 27541 syl2an 465 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
 |-  ( ph  ->  ps )   &    |-  ( ( ph  /\ 
 ch )  ->  th )   &    |-  (
 ( ps  /\  th )  ->  ta )   =>    |-  ( ( ph  /\  ch )  ->  ta )
 
Theoremeel2131 27542 syl2an 465 with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016.)
 |-  (
 ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 th )  ->  ta )   &    |-  (
 ( ch  /\  ta )  ->  et )   =>    |-  ( ( ph  /\ 
 ps  /\  th )  ->  et )
 
Theoremeel3132 27543 syl2an 465 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
 |-  (
 ( ph  /\  ps )  ->  ch )   &    |-  ( ( th  /\ 
 ps )  ->  ta )   &    |-  (
 ( ch  /\  ta )  ->  et )   =>    |-  ( ( ph  /\ 
 th  /\  ps )  ->  et )
 
Theoremeel221 27544 syl2an 465 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
 |-  ( ph  ->  ps )   &    |-  ( ( ch 
 /\  ph )  ->  th )   &    |-  (
 ( ps  /\  th )  ->  ta )   =>    |-  ( ( ch  /\  ph )  ->  ta )
 
Theoremeel0321old 27545 el0321old 27546 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   &    |-  ( ( ph  /\ 
 ta )  ->  et )   =>    |-  (
 ( ps  /\  ch  /\ 
 th )  ->  et )
 
Theoremel0321old 27546 A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. (. ps ,. ch ,. th ).  ->.  ta
 ).   &    |-  ( ( ph  /\  ta )  ->  et )   =>    |-  (. (. ps ,. ch ,. th ).  ->.  et
 ).
 
Theoremeel2122old 27547 el2122old 27548 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps )  ->  ch )   &    |-  ( ps  ->  th )   &    |-  ( ps  ->  ta )   &    |-  ( ( ch 
 /\  th  /\  ta )  ->  et )   =>    |-  ( ( ph  /\  ps )  ->  et )
 
Theoremel2122old 27548 A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (. ph
 ,. ps ).  ->.  ch ).   &    |-  (. ps  ->.  th
 ).   &    |- 
 (. ps  ->.  ta ).   &    |-  ( ( ch 
 /\  th  /\  ta )  ->  et )   =>    |- 
 (. (. ph ,. ps ).  ->.  et ).
 
Theoreme12 27549 A virtual deduction elimination rule. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph ,. ch  ->.  th ).   &    |-  ( ps  ->  ( th  ->  ta ) )   =>    |- 
 (. ph ,. ch  ->.  ta ).
 
Theoreme12an 27550 Conjunction form of e12 27549. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph ,. ch  ->.  th ).   &    |-  (
 ( ps  /\  th )  ->  ta )   =>    |- 
 (. ph ,. ch  ->.  ta ).
 
Theoremel12 27551 Virtual deduction form of syl2an 465. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ta  ->.  ch ).   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (. (. ph
 ,. ta ).  ->.  th ).
 
Theoreme20 27552 A virtual deduction elimination rule. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  th   &    |-  ( ch  ->  ( th  ->  ta )
 )   =>    |- 
 (. ph ,. ps  ->.  ta ).
 
Theoreme20an 27553 Conjunction form of e20 27552. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  th   &    |-  ( ( ch 
 /\  th )  ->  ta )   =>    |-  (. ph ,. ps  ->.  ta ).
 
Theoremee20an 27554 e20an 27553 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  th   &    |-  ( ( ch 
 /\  th )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ta ) )
 
Theoreme21 27555 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph  ->.  th ).   &    |-  ( ch  ->  ( th  ->  ta ) )   =>    |- 
 (. ph ,. ps  ->.  ta ).
 
Theoreme21an 27556 Conjunction form of e21 27555. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph  ->.  th ).   &    |-  (
 ( ch  /\  th )  ->  ta )   =>    |- 
 (. ph ,. ps  ->.  ta ).
 
Theoremee21an 27557 e21an 27556 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  th )   &    |-  ( ( ch 
 /\  th )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ta ) )
 
Theoreme333 27558 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ch  ->.  th ).   &    |-  (. ph ,. ps ,. ch  ->.  ta ).   &    |-  (. ph ,. ps ,. ch  ->.  et ).   &    |-  ( th  ->  ( ta  ->  ( et  ->  ze )
 ) )   =>    |- 
 (. ph ,. ps ,. ch  ->.  ze ).
 
Theoreme33 27559 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ch  ->.  th ).   &    |-  (. ph ,. ps ,. ch  ->.  ta ).   &    |-  ( th  ->  ( ta  ->  et ) )   =>    |- 
 (. ph ,. ps ,. ch  ->.  et ).
 
Theoreme33an 27560 Conjunction form of e33 27559. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ch  ->.  th ).   &    |-  (. ph ,. ps ,. ch  ->.  ta ).   &    |-  (
 ( th  /\  ta )  ->  et )   =>    |- 
 (. ph ,. ps ,. ch  ->.  et ).
 
Theoremee33an 27561 e33an 27560 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )   &    |-  ( ( th  /\  ta )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  et )
 ) )
 
Theoreme3 27562 Meta-connective form of syl8 67. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ch  ->.  th ).   &    |-  ( th  ->  ta )   =>    |- 
 (. ph ,. ps ,. ch  ->.  ta ).
 
Theoreme3bi 27563 Biconditional form of e3 27562. syl8ib 224 is e3bi 27563 without virtual deductions. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ch  ->.  th ).   &    |-  ( th 
 <->  ta )   =>    |- 
 (. ph ,. ps ,. ch  ->.  ta ).
 
Theoreme3bir 27564 Right biconditional form of e3 27562. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ch  ->.  th ).   &    |-  ( ta 
 <-> 
 th )   =>    |- 
 (. ph ,. ps ,. ch  ->.  ta ).
 
Theoreme03 27565 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps ,. ch ,. th  ->.  ta ).   &    |-  ( ph  ->  ( ta  ->  et ) )   =>    |- 
 (. ps ,. ch ,. th  ->.  et ).
 
Theoremee03 27566 e03 27565 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   &    |-  ( ph  ->  ( ta  ->  et ) )   =>    |-  ( ps  ->  ( ch  ->  ( th  ->  et ) ) )
 
Theoreme03an 27567 Conjunction form of e03 27565. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps ,. ch ,. th  ->.  ta ).   &    |-  (
 ( ph  /\  ta )  ->  et )   =>    |- 
 (. ps ,. ch ,. th  ->.  et ).
 
Theoremee03an 27568 Conjunction form of ee03 27566. (Contributed by Alan Sare, 18-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   &    |-  ( ( ph  /\  ta )  ->  et )   =>    |-  ( ps  ->  ( ch  ->  ( th  ->  et ) ) )
 
Theoreme30 27569 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ch  ->.  th ).   &    |-  ta   &    |-  ( th  ->  ( ta  ->  et ) )   =>    |- 
 (. ph ,. ps ,. ch  ->.  et ).
 
Theoremee30 27570 e30 27569 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ta   &    |-  ( th  ->  ( ta  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) )
 
Theoreme30an 27571 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   &    |-  (
 ( th  /\  ta )  ->  et )   =>    |- 
 (. ph ,. ps ,. ch  ->.  et ).
 
Theoremee30an 27572 Conjunction form of ee30 27570. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ta   &    |-  ( ( th  /\ 
 ta )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  et )
 ) )
 
Theoreme13 27573 A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph ,. ch ,. th  ->.  ta ).   &    |-  ( ps  ->  ( ta  ->  et )
 )   =>    |- 
 (. ph ,. ch ,. th  ->.  et ).
 
Theoreme13an 27574 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  ->.  ta ).   &    |-  ( ( ps 
 /\  ta )  ->  et )   =>    |-  (. ph ,. ch ,. th  ->.  et ).
 
Theoremee13an 27575 e13an 27574 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  ( th  ->  ta ) ) )   &    |-  ( ( ps  /\  ta )  ->  et )   =>    |-  ( ph  ->  ( ch  ->  ( th  ->  et )
 ) )
 
Theoreme31 27576 A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ch  ->.  th ).   &    |-  (. ph  ->.  ta
 ).   &    |-  ( th  ->  ( ta  ->  et ) )   =>    |-  (.
 ph ,. ps ,. ch  ->.  et
 ).
 
Theoremee31 27577 e31 27576 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ph  ->  ta )   &    |-  ( th  ->  ( ta  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) )
 
Theoreme31an 27578 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
 ).   &    |-  ( ( th  /\  ta )  ->  et )   =>    |-  (. ph ,. ps ,. ch  ->.  et ).
 
Theoremee31an 27579 e31an 27578 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ph  ->  ta )   &    |-  ( ( th  /\ 
 ta )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  et )
 ) )
 
Theoreme23 27580 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  ->.  ta ).   &    |-  ( ch  ->  ( ta  ->  et ) )   =>    |- 
 (. ph ,. ps ,. th  ->.  et ).
 
Theoreme23an 27581 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  /\  ta )  ->  et )   =>    |-  (. ph ,. ps ,. th  ->.  et ).
 
Theoremee23an 27582 e23an 27581 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  ( th  ->  ta ) ) )   &    |-  ( ( ch  /\  ta )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( th  ->  et )
 ) )
 
Theoreme32 27583 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ch  ->.  th ).   &    |-  (. ph ,. ps  ->.  ta ).   &    |-  ( th  ->  ( ta  ->  et )
 )   =>    |- 
 (. ph ,. ps ,. ch  ->.  et ).
 
Theoremee32 27584 e32 27583 without virtual deductions. (Contributed by Alan Sare, 18-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ph  ->  ( ps  ->  ta )
 )   &    |-  ( th  ->  ( ta  ->  et ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  et )
 ) )
 
Theoreme32an 27585 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 ).   &    |-  ( ( th  /\ 
 ta )  ->  et )   =>    |-  (. ph ,. ps ,. ch  ->.  et ).
 
Theoremee32an 27586 e33an 27560 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ph  ->  ( ps  ->  ta )
 )   &    |-  ( ( th  /\  ta )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  et )
 ) )
 
Theoreme123 27587 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph ,. ch  ->.  th ).   &    |-  (. ph ,. ch ,. ta  ->.  et ).   &    |-  ( ps  ->  ( th  ->  ( et  ->  ze )
 ) )   =>    |- 
 (. ph ,. ch ,. ta  ->.  ze ).
 
Theoremee123 27588 e123 27587 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   &    |-  ( ph  ->  ( ch  ->  ( ta  ->  et ) ) )   &    |-  ( ps  ->  ( th  ->  ( et  ->  ze )
 ) )   =>    |-  ( ph  ->  ( ch  ->  ( ta  ->  ze ) ) )
 
Theoremel123 27589 A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ch  ->.  th ).   &    |-  (. ta  ->.  et ).   &    |-  (
 ( ps  /\  th  /\ 
 et )  ->  ze )   =>    |-  (. (. ph
 ,. ch ,. ta ).  ->.  ze
 ).
 
Theoreme233 27590 A virtual deduction elimination rule. (Contributed by Alan Sare, 29-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph ,. ps ,. th  ->.  ta ).   &    |-  (. ph ,. ps ,. th  ->.  et ).   &    |-  ( ch  ->  ( ta  ->  ( et  ->  ze )
 ) )   =>    |- 
 (. ph ,. ps ,. th  ->.  ze ).
 
Theoreme323 27591 A virtual deduction elimination rule. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ch  ->.  th ).   &    |-  (. ph ,. ps  ->.  ta ).   &    |-  (. ph ,. ps ,. ch  ->.  et ).   &    |-  ( th  ->  ( ta  ->  ( et  ->  ze )
 ) )   =>    |- 
 (. ph ,. ps ,. ch  ->.  ze ).
 
Theoreme000 27592 A virtual deduction elimination rule. The non-virtual deduction form of e000 27592 is the virtual deduction form. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  ch   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  th
 
Theoreme00 27593 Elimination rule identical to mp2 19. The non-virtual deduction form is the virtual deduction form, which is mp2 19. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |- 
 ch
 
Theoreme00an 27594 Elimination rule identical to mp2an 656. The non-virtual deduction form is the virtual deduction form, which is mp2an 656. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  ( ( ph  /\ 
 ps )  ->  ch )   =>    |-  ch
 
Theoremeel00cT 27595 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  ( ( ph  /\ 
 ps )  ->  ch )   =>    |-  (  T.  ->  ch )
 
TheoremeelTT 27596 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  T.  ->  ph )   &    |-  (  T.  ->  ps )   &    |-  ( ( ph  /\ 
 ps )  ->  ch )   =>    |-  ch
 
Theoreme0_ 27597 Elimination rule identical to ax-mp 10. The non-virtual deduction form is the virtual deduction form, which is ax-mp 10. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ph  ->  ps )   =>    |- 
 ps
 
TheoremeelT 27598 An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  T.  ->  ph )   &    |-  ( ph  ->  ps )   =>    |- 
 ps
 
Theoremeel0cT 27599 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ph  ->  ps )   =>    |-  (  T.  ->  ps )
 
TheoremeelT0 27600 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  T.  ->  ph )   &    |-  ps   &    |-  ( ( ph  /\ 
 ps )  ->  ch )   =>    |-  ch
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