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Theorem List for Metamath Proof Explorer - 27601-27700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoreme10an 27601 Conjunction form of e10 27600. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 ch   &    |-  ( ( ps  /\  ch )  ->  th )   =>    |-  (. ph  ->.  th
 ).
 
Theoremee10an 27602 e10an 27601 without virtual deductions. sylancl 646 is also e10an 27601 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 27603 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 27604 Conjunction form of e02 27603. (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 27605 e02an 27604 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 27606 el021old 27607 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 27607 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 27608 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 27609 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 27610 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 27611 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 27612 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 27613 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 27614 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 27615 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 27616 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 27617 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 27618 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 27619 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 27620 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 27621 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 27622 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 27623 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 27624 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 27625 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 27626 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 27627 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 27628 el0321old 27629 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 27629 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 27630 el2122old 27631 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 27631 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 27632 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 27633 Conjunction form of e12 27632. (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 27634 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 27635 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 27636 Conjunction form of e20 27635. (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 27637 e20an 27636 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 27638 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 27639 Conjunction form of e21 27638. (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 27640 e21an 27639 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 27641 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 27642 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 27643 Conjunction form of e33 27642. (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 27644 e33an 27643 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 27645 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 27646 Biconditional form of e3 27645. syl8ib 224 is e3bi 27646 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 27647 Right biconditional form of e3 27645. (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 27648 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 27649 e03 27648 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 27650 Conjunction form of e03 27648. (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 27651 Conjunction form of ee03 27649. (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 27652 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 27653 e30 27652 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 27654 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 27655 Conjunction form of ee30 27653. (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 27656 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 27657 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 27658 e13an 27657 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 27659 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 27660 e31 27659 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 27661 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 27662 e31an 27661 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 27663 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 27664 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 27665 e23an 27664 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 27666 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 27667 e32 27666 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 27668 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 27669 e33an 27643 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 27670 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 27671 e123 27670 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 27672 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 27673 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 27674 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 27675 A virtual deduction elimination rule. The non-virtual deduction form of e000 27675 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 27676 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 27677 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 27678 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 27679 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_ 27680 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 27681 An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  T.  ->  ph )   &    |-  ( ph  ->  ps )   =>    |- 
 ps
 
Theoremeel0cT 27682 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ph  ->  ps )   =>    |-  (  T.  ->  ps )
 
TheoremeelT0 27683 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
 
Theoreme0bi 27684 Elimination rule identical to mpbi 201. The non-virtual deduction form is the virtual deduction form, which is mpbi 201. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ph  <->  ps )   =>    |- 
 ps
 
Theoreme0bir 27685 Elimination rule identical to mpbir 202. The non-virtual deduction form is the virtual deduction form, which is mpbir 202. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  <->  ph )   =>    |- 
 ps
 
Theoremuun0.1 27686 Convention notation form of un0.1 27687. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  T.  ->  ph )   &    |-  ( ps  ->  ch )   &    |-  ( (  T. 
 /\  ps )  ->  th )   =>    |-  ( ps  ->  th )
 
Theoremun0.1 27687  T. is the constant true, a tautology ( see: df-tru 1315). Kleene's "empty conjunction" is logically equivalent to  T.. In a virtual deduction we shall interpret 
T. to be the empty wff or the empty collection of virtual hypotheses.  T. in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If  th is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (.  T.  ->.  ph ).   &    |-  (.
 ps 
 ->.  ch ).   &    |-  (. (.  T.  ,.
 ps ).  ->.  th ).   =>    |- 
 (. ps  ->.  th ).
 
TheoremuunT1 27688 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 (  T.  /\  ph )  ->  ps )   =>    |-  ( ph  ->  ps )
 
TheoremuunT1p1 27689 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  T.  )  ->  ps )   =>    |-  ( ph  ->  ps )
 
TheoremuunT21 27690 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 (  T.  /\  ( ph  /\  ps ) ) 
 ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremuun121 27691 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ( ph  /\ 
 ps ) )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremuun121p1 27692 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  /\  ps )  /\  ph )  ->  ch )   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
Theoremuun132 27693 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ( ps 
 /\  ch ) )  ->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theoremuun132p1 27694 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ps  /\  ch )  /\  ph )  ->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theoremanabss7p1 27695 A deduction unionizing a non-unionized collection of virtual hypotheses. This would have been named uun221 if the 0th permutation did not exist in set.mm as anabss7 797. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ps  /\  ph )  /\  ph )  ->  ch )   =>    |-  ( ( ps  /\  ph )  ->  ch )
 
Theoremun10 27696 A unionizing deduction (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (. ph
 ,.  T.  ).  ->.  ps ).   =>    |-  (. ph  ->.  ps ).
 
Theoremun01 27697 A unionizing deduction (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (.  T.  ,. ph ).  ->.  ps ).   =>    |-  (. ph  ->.  ps ).
 
Theoremun2122 27698 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  /\  ps )  /\  ps  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremuun2131 27699 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  /\  ps )  /\  ( ph  /\  ch ) )  ->  th )   =>    |-  (
 ( ph  /\  ps  /\  ch )  ->  th )
 
Theoremuun2131p1 27700 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  /\  ch )  /\  ( ph  /\  ps ) )  ->  th )   =>    |-  (
 ( ph  /\  ps  /\  ch )  ->  th )
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