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Theorem List for Metamath Proof Explorer - 28501-28600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremel12 28501 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 28502 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 28503 Conjunction form of e20 28502. (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 28504 e20an 28503 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 28505 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 28506 Conjunction form of e21 28505. (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 28507 e21an 28506 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 28508 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 28509 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 28510 Conjunction form of e33 28509. (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 28511 e33an 28510 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 28512 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 28513 Biconditional form of e3 28512. syl8ib 222 is e3bi 28513 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 28514 Right biconditional form of e3 28512. (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 28515 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 28516 e03 28515 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 28517 Conjunction form of e03 28515. (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 28518 Conjunction form of ee03 28516. (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 28519 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 28520 e30 28519 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 28521 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 28522 Conjunction form of ee30 28520. (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 28523 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 28524 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 28525 e13an 28524 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 28526 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 28527 e31 28526 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 28528 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 28529 e31an 28528 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 28530 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 28531 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 28532 e23an 28531 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 28533 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 28534 e32 28533 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 28535 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 28536 e33an 28510 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 28537 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 28538 e123 28537 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 28539 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 28540 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 28541 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 28542 A virtual deduction elimination rule. The non-virtual deduction form of e000 28542 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 28543 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 28544 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 28545 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 28546 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_ 28547 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 28548 An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  T.  ->  ph )   &    |-  ( ph  ->  ps )   =>    |- 
 ps
 
Theoremeel0cT 28549 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ph  ->  ps )   =>    |-  (  T.  ->  ps )
 
TheoremeelT0 28550 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 28551 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 28552 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 28553 Convention notation form of un0.1 28554. (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 28554  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 28555 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 28556 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 28557 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 28558 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 28559 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 28560 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 28561 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 28562 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 28563 A unionizing deduction (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (. ph
 ,.  T.  ).  ->.  ps ).   =>    |-  (. ph  ->.  ps ).
 
Theoremun01 28564 A unionizing deduction (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (.  T.  ,. ph ).  ->.  ps ).   =>    |-  (. ph  ->.  ps ).
 
Theoremun2122 28565 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 28566 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 28567 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 28568 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 28569 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 28570 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 28571 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 28572 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 28573 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 28574 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 28575 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 28576 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 28577 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 28578 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 28579 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 28580 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 28581 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 28582 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 28583 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 28584 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 28585 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 28586 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 28587 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 28588 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 28589 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 28590 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 28591 A deduction unionizing a non-unionized collection of virtual hypotheses. 3impdir 1238 is ~? uun3132 and is in set.mm. 3impdirp1 28591 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 )
 
18.25.4  Theorems proved using virtual deduction
 
TheoremtrsspwALT 28592 Virtual deduction proof of the left-to-right implication of dftr4 4118. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4118 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 28593 Virtual deduction proof of trsspwALT 28592. This proof is the same as the proof of trsspwALT 28592 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 28594 Short predicate calculus proof of the left-to-right implication of dftr4 4118. 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 28593, which is the virtual deduction proof trsspwALT 28592 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 28595 Virtual deduction proof of the right-to-left implication of dftr4 4118. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 28595 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 28596 Virtual deduction proof of sspwtr 28595. This proof is the same as the proof of sspwtr 28595 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 28597 Short predicate calculus proof of the right-to-left implication of dftr4 4118. 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 28596, which is the virtual deduction proof sspwtr 28595 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 28598 Virtual deduction proof of pwtrOLD 28599. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  Tr  ~P A )
 
TheorempwtrOLD 28599 The power class of a transitive class is transitive. The proof of this theorem was automatically generated from pwtrVD 28598 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 4226 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 28600 Virtual deduction proof of pwtrrOLD 28601. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( Tr  ~P A  ->  Tr  A )
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