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Theorem List for Metamath Proof Explorer - 27201-27300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremee33an 27201 e33an 27200 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 27202 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 27203 Biconditional form of e3 27202. syl8ib 224 is e3bi 27203 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 27204 Right biconditional form of e3 27202. (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 27205 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 27206 e03 27205 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 27207 Conjunction form of e03 27205. (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 27208 Conjunction form of ee03 27206. (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 27209 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 27210 e30 27209 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 27211 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 27212 Conjunction form of ee30 27210. (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 27213 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 27214 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 27215 e13an 27214 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 27216 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 27217 e31 27216 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 27218 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 27219 e31an 27218 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 27220 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 27221 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 27222 e23an 27221 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 27223 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 27224 e32 27223 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 27225 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 27226 e33an 27200 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 27227 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 27228 e123 27227 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 27229 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 27230 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 27231 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 27232 A virtual deduction elimination rule. The non-virtual deduction form of e000 27232 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 27233 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 27234 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 27235 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 27236 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_ 27237 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 27238 An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  T.  ->  ph )   &    |-  ( ph  ->  ps )   =>    |- 
 ps
 
Theoremeel0cT 27239 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ph  ->  ps )   =>    |-  (  T.  ->  ps )
 
TheoremeelT0 27240 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 27241 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 27242 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 27243 Convention notation form of un0.1 27244. (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 27244  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 27245 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 27246 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 27247 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 27248 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 27249 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 27250 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 27251 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 27252 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 27253 A unionizing deduction (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (. ph
 ,.  T.  ).  ->.  ps ).   =>    |-  (. ph  ->.  ps ).
 
Theoremun01 27254 A unionizing deduction (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (.  T.  ,. ph ).  ->.  ps ).   =>    |-  (. ph  ->.  ps ).
 
Theoremun2122 27255 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 27256 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 27257 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 27258 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 27259 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 27260 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 27261 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 27262 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 27263 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 27264 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 27265 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 27266 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 27267 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 27268 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 27269 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 27270 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 27271 A deduction unionizing a non-unionized collection of virtual hypotheses. 3anidm12 1244 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 27272 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 27273 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 27274 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 27275 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 27276 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 )
 
Theoremuun123p4 27277 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  /\  ps  /\  ph )  ->  th )   =>    |-  (
 ( ph  /\  ps  /\  ch )  ->  th )
 
Theoremuun2221 27278 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 30-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ph  /\  ( ps  /\  ph ) )  ->  ch )   =>    |-  ( ( ps  /\  ph )  ->  ch )
 
Theoremuun2221p1 27279 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 )  /\  ph )  ->  ch )   =>    |-  ( ( ps  /\  ph )  ->  ch )
 
Theoremuun2221p2 27280 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  /\  ph )  ->  ch )   =>    |-  ( ( ps  /\  ph )  ->  ch )
 
Theorem3impdirp1 27281 A deduction unionizing a non-unionized collection of virtual hypotheses. 3impdir 1243 is ~? uun3132 and is in set.mm. 3impdirp1 27281 is ~? uun3132p1. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ch  /\  ps )  /\  ( ph  /\ 
 ps ) )  ->  th )   =>    |-  ( ( ph  /\  ch  /\ 
 ps )  ->  th )
 
16.20.4  Theorems proved using virtual deduction
 
TheoremtrsspwALT 27282 Virtual deduction proof of the left-to-right implication of dftr4 4015. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4015 without accumulating results. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  A  C_  ~P A )
 
TheoremtrsspwALT2 27283 Virtual deduction proof of trsspwALT 27282. This proof is the same as the proof of trsspwALT 27282 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A transitive class is a subset of its power class. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  A  C_  ~P A )
 
TheoremtrsspwALT3 27284 Short predicate calculus proof of the left-to-right implication of dftr4 4015. A transitive class is a subset of its power class. This proof was constructed by applying Metamath's minimize command to the proof of trsspwALT2 27283, which is the virtual deduction proof trsspwALT 27282 without virtual deductions. (Contributed by Alan Sare, 30-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  A  C_  ~P A )
 
Theoremsspwtr 27285 Virtual deduction proof of the right-to-left implication of dftr4 4015. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 27285 without accumulating results. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  C_  ~P A  ->  Tr  A )
 
TheoremsspwtrALT 27286 Virtual deduction proof of sspwtr 27285. This proof is the same as the proof of sspwtr 27285 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A class which is a subclass of its power class is transitive. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  C_  ~P A  ->  Tr  A )
 
TheoremsspwtrALT2 27287 Short predicate calculus proof of the right-to-left implication of dftr4 4015. A class which is a subclass of its power class is transitive. This proof was constructed by applying Metamath's minimize command to the proof of sspwtrALT 27286, which is the virtual deduction proof sspwtr 27285 without virtual deductions. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  C_  ~P A  ->  Tr  A )
 
TheorempwtrVD 27288 Virtual deduction proof of pwtrOLD 27289. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  Tr  ~P A )
 
TheorempwtrOLD 27289 The power class of a transitive class is transitive. The proof of this theorem was automatically generated from pwtrVD 27288 using a tools command file, translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Moved into main set.mm as pwtr 4120 and may be deleted by mathbox owner, AS. --NM 15-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  Tr  ~P A )
 
TheorempwtrrVD 27290 Virtual deduction proof of pwtrrOLD 27291. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( Tr  ~P A  ->  Tr  A )
 
TheorempwtrrOLD 27291 A set is transitive if its power set is. The proof of this theorem was automatically generated from pwtrrVD 27290 using a tools command file, translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Moved into main set.mm as pwtr 4120 and may be deleted by mathbox owner, AS. --NM 15-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( Tr  ~P A  ->  Tr  A )
 
TheoremsnssiALTVD 27292 Virtual deduction proof of snssiALT 27293. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  { A }  C_  B )
 
TheoremsnssiALT 27293 If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 3659. This theorem was automatically generated from snssiALTVD 27292 using a translation program. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  { A }  C_  B )
 
TheoremsnsslVD 27294 Virtual deduction proof of snssl 27295. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( { A }  C_  B  ->  A  e.  B )
 
Theoremsnssl 27295 If a singleton is a subclass of another class, then the singleton's element is an element of that other class. This theorem is the right-to-left implication of the biconditional snss 3652. The proof of this theorem was automatically generated from snsslVD 27294 using a tools command file, translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( { A }  C_  B  ->  A  e.  B )
 
TheoremsnelpwrVD 27296 Virtual deduction proof of snelpwi 4114. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  { A }  e.  ~P B )
 
TheoremsnelpwrOLD 27297 If a class is contained in another class, then its singleton is contained in the power class of that other class. This theorem is the left-to-right implication of the biconditional snelpw 4115. Unlike snelpw 4115, 
A may be a proper class. The proof of this theorem was automatically generated from snelpwrVD 27296 using a tools command file, translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Moved to snelpwi 4114 in main set.mm and may be deleted by mathbox owner, AS. --NM 10-Sep-2013.) (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  { A }  e.  ~P B )
 
TheoremunipwrVD 27298 Virtual deduction proof of unipwr 27299. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  C_ 
 U. ~P A
 
Theoremunipwr 27299 A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 4118. The proof of this theorem was automatically generated from unipwrVD 27298 using a tools command file , translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  C_ 
 U. ~P A
 
TheoremsstrALT2VD 27300 Virtual deduction proof of sstrALT2 27301. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
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