HomeHome Metamath Proof Explorer
Theorem List (p. 272 of 310)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-30955)
 

Theorem List for Metamath Proof Explorer - 27101-27200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoreme202 27101 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  th   &    |-  (. ph ,. ps  ->.  ta ).   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. ps  ->.  et ).
 
Theoremee202 27102 e202 27101 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  th   &    |-  ( ph  ->  ( ps  ->  ta )
 )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et )
 )
 
Theoreme022 27103 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps ,. ch  ->. 
 th ).   &    |-  (. ps ,. ch  ->.  ta ).   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ps ,. ch  ->.  et ).
 
Theoremee022 27104 e022 27103 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ( ch  ->  th )
 )   &    |-  ( ps  ->  ( ch  ->  ta ) )   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ps  ->  ( ch  ->  et ) )
 
Theoreme002 27105 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  (. ch ,. th  ->.  ta ).   &    |-  ( ph  ->  ( ps  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ch ,. th  ->.  et ).
 
Theoremee002 27106 e002 27105 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  ( ch  ->  ( th  ->  ta )
 )   &    |-  ( ph  ->  ( ps  ->  ( ta  ->  et ) ) )   =>    |-  ( ch  ->  ( th  ->  et )
 )
 
Theoreme020 27107 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps ,. ch  ->. 
 th ).   &    |-  ta   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ps ,. ch  ->.  et ).
 
Theoremee020 27108 e020 27107 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ( ch  ->  th )
 )   &    |- 
 ta   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ps  ->  ( ch  ->  et )
 )
 
Theoreme200 27109 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  th   &    |-  ta   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. ps  ->.  et ).
 
Theoremee200 27110 e200 27109 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  th   &    |-  ta   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoreme221 27111 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph ,. ps  ->. 
 th ).   &    |-  (. ph  ->.  ta ).   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ph ,. ps  ->.  et ).
 
Theoremee221 27112 e221 27111 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th )
 )   &    |-  ( ph  ->  ta )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoreme212 27113 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph  ->.  th ).   &    |-  (. ph ,. ps  ->.  ta ).   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. ps  ->.  et ).
 
Theoremee212 27114 e212 27113 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ( ps  ->  ta )
 )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et )
 )
 
Theoreme122 27115 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph ,. ch  ->.  th ).   &    |-  (. ph ,. ch  ->.  ta ).   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. ch  ->.  et ).
 
Theoreme112 27116 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph  ->.  ch ).   &    |-  (. ph ,. th  ->.  ta ).   &    |-  ( ps  ->  ( ch  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. th  ->.  et ).
 
Theoremee112 27117 e112 27116 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( th  ->  ta )
 )   &    |-  ( ps  ->  ( ch  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( th  ->  et )
 )
 
Theoreme121 27118 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph ,. ch  ->.  th ).   &    |-  (. ph  ->.  ta
 ).   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (. ph ,. ch  ->.  et ).
 
Theoreme211 27119 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph  ->.  th ).   &    |-  (. ph  ->.  ta
 ).   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (. ph ,. ps  ->.  et ).
 
Theoremee211 27120 e211 27119 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoreme210 27121 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph  ->.  th ).   &    |-  ta   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ph ,. ps  ->.  et ).
 
Theoremee210 27122 e210 27121 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  th )   &    |-  ta   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoreme201 27123 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  th   &    |-  (. ph  ->.  ta ).   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ph ,. ps  ->.  et ).
 
Theoremee201 27124 e201 27123 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  th   &    |-  ( ph  ->  ta )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoreme120 27125 A virtual deduction elimination rule. (Contributed by Alan Sare, 10-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph ,. ch  ->.  th ).   &    |-  ta   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ph ,. ch  ->.  et ).
 
Theoremee120 27126 Virtual deduction rule e120 27125 without virtual deduction symbols. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   &    |- 
 ta   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ch  ->  et )
 )
 
Theoreme021 27127 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps ,. ch  ->. 
 th ).   &    |-  (. ps  ->.  ta ).   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ps ,. ch  ->.  et ).
 
Theoremee021 27128 e021 27127 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ( ch  ->  th )
 )   &    |-  ( ps  ->  ta )   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ps  ->  ( ch  ->  et ) )
 
Theoreme012 27129 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps  ->.  ch ).   &    |-  (. ps ,. th  ->.  ta ).   &    |-  ( ph  ->  ( ch  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ps ,. th  ->.  et ).
 
Theoremee012 27130 e012 27129 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  ( ps  ->  ( th  ->  ta )
 )   &    |-  ( ph  ->  ( ch  ->  ( ta  ->  et ) ) )   =>    |-  ( ps  ->  ( th  ->  et )
 )
 
Theoreme102 27131 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 ch   &    |- 
 (. ph ,. th  ->.  ta ).   &    |-  ( ps  ->  ( ch  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ph ,. th  ->.  et ).
 
Theoremee102 27132 e102 27131 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  ( ph  ->  ( th  ->  ta )
 )   &    |-  ( ps  ->  ( ch  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( th  ->  et )
 )
 
Theoreme22 27133 A virtual deduction elimination rule. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph ,. ps  ->. 
 th ).   &    |-  ( ch  ->  ( th  ->  ta )
 )   =>    |- 
 (. ph ,. ps  ->.  ta ).
 
Theoreme22an 27134 Conjunction form of e22 27133. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph ,. ps  ->. 
 th ).   &    |-  ( ( ch 
 /\  th )  ->  ta )   =>    |-  (. ph ,. ps  ->.  ta ).
 
Theoremee22an 27135 e22an 27134 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th )
 )   &    |-  ( ( ch  /\  th )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ta ) )
 
Theoreme111 27136 A virtual deduction elimination rule. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph  ->.  ch ).   &    |-  (. ph  ->.  th ).   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) )   =>    |- 
 (. ph  ->.  ta ).
 
Theoreme1111 27137 A virtual deduction elimination rule. (Contributed by Alan Sare, 6-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph  ->.  ch ).   &    |-  (. ph  ->.  th ).   &    |-  (. ph  ->.  ta
 ).   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) )   =>    |-  (. ph  ->.  et ).
 
Theoreme110 27138 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph  ->.  ch ).   &    |-  th   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  (.
 ph 
 ->.  ta ).
 
Theoremee110 27139 e110 27138 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  th   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  ( ph  ->  ta )
 
Theoreme101 27140 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 ch   &    |- 
 (. ph  ->.  th ).   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  (.
 ph 
 ->.  ta ).
 
Theoremee101 27141 e101 27140 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 27142 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 27143 e011 27142 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 27144 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 27145 e100 27144 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 27146 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 27147 e010 27146 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 27148 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 27149 e001 27148 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 27150 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 27151 Conjunction form of e11 27150. (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 27152 e11an 27151 without virtual deductions. syl22anc 1188 is also e11an 27151 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 27153 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 27154 e01 27153 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 27155 Conjunction form of e01 27153. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps  ->.  ch ).   &    |-  (
 ( ph  /\  ch )  ->  th )   =>    |- 
 (. ps  ->.  th ).
 
Theoremee01an 27156 e01an 27155 without virtual deductions. sylancr 647 is also a form of e01an 27155 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 27157 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 27158 Conjunction form of e10 27157. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 ch   &    |-  ( ( ps  /\  ch )  ->  th )   =>    |-  (. ph  ->.  th
 ).
 
Theoremee10an 27159 e10an 27158 without virtual deductions. sylancl 646 is also e10an 27158 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 27160 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 27161 Conjunction form of e02 27160. (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 27162 e02an 27161 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 27163 el021old 27164 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 27164 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 27165 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 27166 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 27167 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 27168 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 27169 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 27170 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 27171 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 27172 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 27173 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 27174 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 27175 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 27176 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 27177 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 27178 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 27179 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 27180 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 27181 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 27182 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 27183 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 27184 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 27185 el0321old 27186 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 27186 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 27187 el2122old 27188 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 27188 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 27189 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 27190 Conjunction form of e12 27189. (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 27191 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 27192 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 27193 Conjunction form of e20 27192. (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 27194 e20an 27193 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 27195 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 27196 Conjunction form of e21 27195. (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 27197 e21an 27196 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 27198 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 27199 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 27200 Conjunction form of e33 27199. (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 ).
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-30955
  Copyright terms: Public domain < Previous  Next >