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Theorem List for Metamath Proof Explorer - 28801-28900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeel32131 28801 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   &    |-  (
 ( ph  /\  ch )  ->  ta )   &    |-  ( ( th  /\ 
 ta )  ->  et )   =>    |-  (
 ( ph  /\  ps  /\  ch )  ->  et )
 
Theoremeel221 28802 syl2an 463 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 28803 el0321old 28804 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 28804 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 28805 el2122old 28806 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 28806 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 ).
 
Theoremeel0001 28807 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  ph   &    |-  ps   &    |-  ch   &    |-  ( th  ->  ta )   &    |-  ( ( ( ( ph  /\  ps )  /\  ch )  /\  ta )  ->  et )   =>    |-  ( th  ->  et )
 
Theoremeel0000 28808 Elimination rule similar to mp4an 654, except with a left-nested conjunction unification theorem. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  ph   &    |-  ps   &    |-  ch   &    |-  th   &    |-  ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  ->  ta )   =>    |-  ta
 
Theoremeel1111 28809 4 hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl112anc 1186 except the unification theorem uses left-nested conjunction. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ( ( ( ps  /\  ch )  /\  th )  /\  ta )  ->  et )   =>    |-  ( ph  ->  et )
 
Theoremeel00001 28810 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  ph   &    |-  ps   &    |-  ch   &    |-  th   &    |-  ( ta  ->  et )   &    |-  ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  et )  ->  ze )   =>    |-  ( ta  ->  ze )
 
Theoremeel00000 28811 Elimination rule similar eel0000 28808, except with five hpothesis steps (Contributed by Alan Sare, 17-Oct-2017.)
 |-  ph   &    |-  ps   &    |-  ch   &    |-  th   &    |-  ta   &    |-  ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  ->  et )   =>    |- 
 et
 
Theoremeel11111 28812 5 hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl113anc 1194 except the unification theorem uses left-nested conjunction. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ( ( ( ( ps  /\  ch )  /\  th )  /\  ta )  /\  et )  ->  ze )   =>    |-  ( ph  ->  ze )
 
Theoreme12 28813 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 28814 Conjunction form of e12 28813. (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 28815 Virtual deduction form of syl2an 463. (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 28816 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 28817 Conjunction form of e20 28816. (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 28818 e20an 28817 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 28819 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 28820 Conjunction form of e21 28819. (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 28821 e21an 28820 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 28822 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 28823 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 28824 Conjunction form of e33 28823. (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 28825 e33an 28824 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 28826 Meta-connective form of syl8 65. (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 28827 Biconditional form of e3 28826. syl8ib 222 is e3bi 28827 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 28828 Right biconditional form of e3 28826. (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 28829 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 28830 e03 28829 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 28831 Conjunction form of e03 28829. (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 28832 Conjunction form of ee03 28830. (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 28833 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 28834 e30 28833 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 28835 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 28836 Conjunction form of ee30 28834. (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 28837 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 28838 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 28839 e13an 28838 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 28840 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 28841 e31 28840 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 28842 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 28843 e31an 28842 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 28844 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 28845 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 28846 e23an 28845 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 28847 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 28848 e32 28847 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 28849 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 28850 e33an 28824 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 28851 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 28852 e123 28851 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 28853 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 28854 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 28855 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 28856 A virtual deduction elimination rule. The non-virtual deduction form of e000 28856 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 28857 Elimination rule identical to mp2 17. The non-virtual deduction form is the virtual deduction form, which is mp2 17. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |- 
 ch
 
Theoreme00an 28858 Elimination rule identical to mp2an 653. The non-virtual deduction form is the virtual deduction form, which is mp2an 653. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  ( ( ph  /\ 
 ps )  ->  ch )   =>    |-  ch
 
Theoremeel00cT 28859 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 28860 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_ 28861 Elimination rule identical to ax-mp 8. The non-virtual deduction form is the virtual deduction form, which is ax-mp 8. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ph  ->  ps )   =>    |- 
 ps
 
TheoremeelT 28862 An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  T.  ->  ph )   &    |-  ( ph  ->  ps )   =>    |- 
 ps
 
Theoremeel0cT 28863 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ph  ->  ps )   =>    |-  (  T.  ->  ps )
 
TheoremeelT0 28864 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 28865 Elimination rule identical to mpbi 199. The non-virtual deduction form is the virtual deduction form, which is mpbi 199. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ph  <->  ps )   =>    |- 
 ps
 
Theoreme0bir 28866 Elimination rule identical to mpbir 200. The non-virtual deduction form is the virtual deduction form, which is mpbir 200. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  <->  ph )   =>    |- 
 ps
 
Theoremuun0.1 28867 Convention notation form of un0.1 28868. (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 28868  T. is the constant true, a tautology ( see: df-tru 1310). 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 28869 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 28870 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 28871 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 28872 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 28873 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 28874 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 28875 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 28876 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 794. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ps  /\  ph )  /\  ph )  ->  ch )   =>    |-  ( ( ps  /\  ph )  ->  ch )
 
Theoremun10 28877 A unionizing deduction (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (. ph
 ,.  T.  ).  ->.  ps ).   =>    |-  (. ph  ->.  ps ).
 
Theoremun01 28878 A unionizing deduction (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (.  T.  ,. ph ).  ->.  ps ).   =>    |-  (. ph  ->.  ps ).
 
Theoremun2122 28879 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 28880 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 28881 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 )
 
TheoremuunTT1 28882 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.)
 |-  (
 (  T.  /\  T.  /\  ph )  ->  ps )   =>    |-  ( ph  ->  ps )
 
TheoremuunTT1p1 28883 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.)
 |-  (
 (  T.  /\  ph  /\  T.  )  ->  ps )   =>    |-  ( ph  ->  ps )
 
TheoremuunTT1p2 28884 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.  /\  T.  )  ->  ps )   =>    |-  ( ph  ->  ps )
 
TheoremuunT11 28885 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.)
 |-  (
 (  T.  /\  ph  /\  ph )  ->  ps )   =>    |-  ( ph  ->  ps )
 
TheoremuunT11p1 28886 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.  /\  ph )  ->  ps )   =>    |-  ( ph  ->  ps )
 
TheoremuunT11p2 28887 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  /\  T.  )  ->  ps )   =>    |-  ( ph  ->  ps )
 
TheoremuunT12 28888 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.)
 |-  (
 (  T.  /\  ph  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
TheoremuunT12p1 28889 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.)
 |-  (
 (  T.  /\  ps  /\  ph )  ->  ch )   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
TheoremuunT12p2 28890 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 )  ->  ch )   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
TheoremuunT12p3 28891 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  /\  T.  /\  ph )  ->  ch )   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
TheoremuunT12p4 28892 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  /\  T.  )  ->  ch )   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
TheoremuunT12p5 28893 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  /\  ph  /\  T.  )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremuun111 28894 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  /\  ph )  ->  ps )   =>    |-  ( ph  ->  ps )
 
Theorem3anidm12p1 28895 A deduction unionizing a non-unionized collection of virtual hypotheses. 3anidm12 1239 denotes the deduction which would have been named uun112 if it did not pre-exist in set.mm. This second permutation's name is based on this pre-existing name. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ph )  ->  ch )   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
Theorem3anidm12p2 28896 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  /\  ph  /\  ph )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremuun123 28897 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  /\  ps )  ->  th )   =>    |-  (
 ( ph  /\  ps  /\  ch )  ->  th )
 
Theoremuun123p1 28898 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  /\  ph  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theoremuun123p2 28899 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.)
 |-  (
 ( ch  /\  ph  /\  ps )  ->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theoremuun123p3 28900 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 )
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