HomeHome Metamath Proof Explorer
Theorem List (p. 284 of 323)
< 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
(1-21612)
  Hilbert Space Explorer  Hilbert Space Explorer
(21613-23135)
  Users' Mathboxes  Users' Mathboxes
(23136-32223)
 

Theorem List for Metamath Proof Explorer - 28301-28400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoreme323 28301 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 28302 A virtual deduction elimination rule. The non-virtual deduction form of e000 28302 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 28303 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 28304 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 28305 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 28306 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_ 28307 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 28308 An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (  T.  ->  ph )   &    |-  ( ph  ->  ps )   =>    |- 
 ps
 
Theoremeel0cT 28309 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ph  ->  ps )   =>    |-  (  T.  ->  ps )
 
TheoremeelT0 28310 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 28311 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 28312 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 28313 Convention notation form of un0.1 28314. (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 28314  T. is the constant true, a tautology ( see: df-tru 1319). 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 28315 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 28316 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 28317 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 28318 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 28319 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 28320 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 28321 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 28322 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 28323 A unionizing deduction (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (. ph
 ,.  T.  ).  ->.  ps ).   =>    |-  (. ph  ->.  ps ).
 
Theoremun01 28324 A unionizing deduction (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (.  T.  ,. ph ).  ->.  ps ).   =>    |-  (. ph  ->.  ps ).
 
Theoremun2122 28325 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 28326 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 28327 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 28328 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 28329 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 28330 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 28331 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 28332 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 28333 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 28334 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 28335 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 28336 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 28337 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 28338 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 28339 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 28340 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 28341 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 28342 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 28343 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 28344 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 28345 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 28346 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 28347 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 28348 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 28349 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 28350 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 28351 A deduction unionizing a non-unionized collection of virtual hypotheses. 3impdir 1238 is ~? uun3132 and is in set.mm. 3impdirp1 28351 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 )
 
Theorem3impcombi 28352 A 1-hypothesis propositional calculus deduction (Contributed by Alan Sare, 25-Sep-2017.)
 |-  (
 ( ph  /\  ps  /\  ph )  ->  ( ch  <->  th ) )   =>    |-  ( ( ps  /\  ph 
 /\  ch )  ->  th )
 
Theorem3imp231 28353 Importation inference. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   =>    |-  ( ( ps  /\  ch 
 /\  ph )  ->  th )
 
18.24.6  Theorems proved using virtual deduction
 
TheoremtrsspwALT 28354 Virtual deduction proof of the left-to-right implication of dftr4 4199. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4199 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 28355 Virtual deduction proof of trsspwALT 28354. This proof is the same as the proof of trsspwALT 28354 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 28356 Short predicate calculus proof of the left-to-right implication of dftr4 4199. 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 28355, which is the virtual deduction proof trsspwALT 28354 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 28357 Virtual deduction proof of the right-to-left implication of dftr4 4199. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 28357 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 28358 Virtual deduction proof of sspwtr 28357. This proof is the same as the proof of sspwtr 28357 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 28359 Short predicate calculus proof of the right-to-left implication of dftr4 4199. 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 28358, which is the virtual deduction proof sspwtr 28357 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 28360 Virtual deduction proof of pwtrOLD 28361. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  Tr  ~P A )
 
TheorempwtrOLD 28361 The power class of a transitive class is transitive. The proof of this theorem was automatically generated from pwtrVD 28360 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 4308 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 28362 Virtual deduction proof of pwtrrOLD 28363. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( Tr  ~P A  ->  Tr  A )
 
TheorempwtrrOLD 28363 A set is transitive if its power set is. The proof of this theorem was automatically generated from pwtrrVD 28362 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 4308 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 28364 Virtual deduction proof of snssiALT 28365. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  { A }  C_  B )
 
TheoremsnssiALT 28365 If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 3838. This theorem was automatically generated from snssiALTVD 28364 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 28366 Virtual deduction proof of snssl 28367. (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 28367 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 3824. The proof of this theorem was automatically generated from snsslVD 28366 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 28368 Virtual deduction proof of snelpwi 4301. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  { A }  e.  ~P B )
 
TheoremsnelpwrOLD 28369 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 4302. Unlike snelpw 4302, 
A may be a proper class. The proof of this theorem was automatically generated from snelpwrVD 28368 using a tools command file, translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Moved to snelpwi 4301 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 28370 Virtual deduction proof of unipwr 28371. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  C_ 
 U. ~P A
 
Theoremunipwr 28371 A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 4306. The proof of this theorem was automatically generated from unipwrVD 28370 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 28372 Virtual deduction proof of sstrALT2 28373. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
 
TheoremsstrALT2 28373 Virtual deduction proof of sstr 3263, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 28372 using the command file translatewithout_overwriting.cmd . It was not minimized because the automated minimization excluding duplicates generates a minimized proof which, although not directly containing any duplicates, indirectly contains a duplicate. That is, the trace back of the minimized proof contains a duplicate. This is undesirable because some step(s) of the minimized proof use the proven theorem. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
 
TheoremsuctrALT2VD 28374 Virtual deduction proof of suctrALT2 28375. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  Tr  suc  A )
 
TheoremsuctrALT2 28375 Virtual deduction proof of suctr 4557. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 28374 using the tools command file translatewithout_overwritingminimize_excludingduplicates.cmd . (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  Tr  suc  A )
 
Theoremelex2VD 28376* Virtual deduction proof of elex2 2876. (Contributed by Alan Sare, 25-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  E. x  x  e.  B )
 
Theoremelex22VD 28377* Virtual deduction proof of elex22 2875. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  B  /\  A  e.  C ) 
 ->  E. x ( x  e.  B  /\  x  e.  C ) )
 
Theoremeqsbc3rVD 28378* Virtual deduction proof of eqsbc3r 3124. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 [. A  /  x ]. C  =  x  <->  C  =  A ) )
 
Theoremzfregs2VD 28379* Virtual deduction proof of zfregs2 7505. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  =/=  (/)  ->  -.  A. x  e.  A  E. y ( y  e.  A  /\  y  e.  x )
 )
 
Theoremtpid3gVD 28380 Virtual deduction proof of tpid3g 3817. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A }
 )
 
Theoremen3lplem1VD 28381* Virtual deduction proof of en3lplem1 7506. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  A  ->  E. y ( y  e.  { A ,  B ,  C }  /\  y  e.  x ) ) )
 
Theoremen3lplem2VD 28382* Virtual deduction proof of en3lplem2 7507. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  E. y
 ( y  e.  { A ,  B ,  C }  /\  y  e.  x ) ) )
 
Theoremen3lpVD 28383 Virtual deduction proof of en3lp 7508. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A )
 
18.24.7  Theorems proved using virtual deduction with mmj2 assistance
 
Theoremsimplbi2VD 28384 Virtual deduction proof of simplbi2 608. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
h1::  |-  ( ph  <->  ( ps  /\  ch ) )
3:1,?: e0_ 28307  |-  ( ( ps  /\  ch )  ->  ph )
qed:3,?: e0_ 28307  |-  ( ps  ->  ( ch  ->  ph ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( ps  /\  ch ) )   =>    |-  ( ps  ->  ( ch  ->  ph ) )
 
Theorem3ornot23VD 28385 Virtual deduction proof of 3ornot23 28015. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::
 |-  (. ( -.  ph  /\  -.  ps )  ->.  ( -.  ph  /\  -.  ps ) ).
2::  |-  (. ( -.  ph  /\  -.  ps ) ,. ( ch  \/  ph  \/  ps )  ->.  ( ch  \/  ph  \/  ps ) ).
3:1,?: e1_ 28150  |-  (. ( -.  ph  /\  -.  ps )  ->.  -.  ph ).
4:1,?: e1_ 28150  |-  (. ( -.  ph  /\  -.  ps )  ->.  -.  ps ).
5:3,4,?: e11 28211  |-  (. ( -.  ph  /\  -.  ps )  ->.  -.  ( ph  \/  ps ) ).
6:2,?: e2 28154  |-  (. ( -.  ph  /\  -.  ps ) ,. ( ch  \/  ph  \/  ps )  ->.  ( ch  \/  ( ph  \/  ps ) ) ).
7:5,6,?: e12 28259  |-  (. ( -.  ph  /\  -.  ps ) ,. ( ch  \/  ph  \/  ps )  ->.  ch ).
8:7:  |-  (. ( -.  ph  /\  -.  ps )  ->.  ( ( ch  \/  ph  \/  ps )  ->  ch ) ).
qed:8:  |-  ( ( -.  ph  /\  -.  ps )  ->  ( ( ch  \/  ph  \/  ps )  ->  ch ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( -.  ph  /\  -.  ps )  ->  ( ( ch  \/  ph  \/  ps )  ->  ch ) )
 
Theoremorbi1rVD 28386 Virtual deduction proof of orbi1r 28016. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ph  <->  ps )  ->.  ( ph  <->  ps ) ).
2::  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ch  \/  ph ) ).
3:2,?: e2 28154  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ph  \/  ch ) ).
4:1,3,?: e12 28259  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ps  \/  ch ) ).
5:4,?: e2 28154  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ch  \/  ps ) ).
6:5:  |-  (. ( ph  <->  ps )  ->.  ( ( ch  \/  ph )  ->  ( ch  \/  ps ) ) ).
7::  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ch  \/  ps ) ).
8:7,?: e2 28154  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ps  \/  ch ) ).
9:1,8,?: e12 28259  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ph  \/  ch ) ).
10:9,?: e2 28154  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ch  \/  ph ) ).
11:10:  |-  (. ( ph  <->  ps )  ->.  ( ( ch  \/  ps )  ->  ( ch  \/  ph ) ) ).
12:6,11,?: e11 28211  |-  (. ( ph  <->  ps )  ->.  ( ( ch  \/  ph )  <->  ( ch  \/  ps ) ) ).
qed:12:  |-  ( ( ph  <->  ps )  ->  ( ( ch  \/  ph )  <->  ( ch  \/  ps ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ch  \/  ph ) 
 <->  ( ch  \/  ps ) ) )
 
Theorembitr3VD 28387 Virtual deduction proof of bitr3 28017. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ph  <->  ps )  ->.  ( ph  <->  ps ) ).
2:1,?: e1_ 28150  |-  (. ( ph  <->  ps )  ->.  ( ps  <->  ph ) ).
3::  |-  (. ( ph  <->  ps ) ,. ( ph  <->  ch )  ->.  ( ph  <->  ch ) ).
4:3,?: e2 28154  |-  (. ( ph  <->  ps ) ,. ( ph  <->  ch )  ->.  ( ch  <->  ph ) ).
5:2,4,?: e12 28259  |-  (. ( ph  <->  ps ) ,. ( ph  <->  ch )  ->.  ( ps  <->  ch ) ).
6:5:  |-  (. ( ph  <->  ps )  ->.  ( ( ph  <->  ch )  ->  ( ps  <->  ch ) ) ).
qed:6:  |-  ( ( ph  <->  ps )  ->  ( ( ph  <->  ch )  ->  ( ps  <->  ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ph  <->  ch )  ->  ( ps 
 <->  ch ) ) )
 
Theorem3orbi123VD 28388 Virtual deduction proof of 3orbi123 28018. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) ) ).
2:1,?: e1_ 28150  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ph  <->  ps ) ).
3:1,?: e1_ 28150  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ch  <->  th ) ).
4:1,?: e1_ 28150  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ta  <->  et ) ).
5:2,3,?: e11 28211  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  \/  ch )  <->  ( ps  \/  th ) ) ).
6:5,4,?: e11 28211  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ( ph  \/  ch )  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) ) ).
7:?:  |-  ( ( ( ph  \/  ch )  \/  ta )  <->  ( ph  \/  ch  \/  ta ) )
8:6,7,?: e10 28218  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  \/  ch  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) ) ).
9:?:  |-  ( ( ( ps  \/  th )  \/  et )  <->  ( ps  \/  th  \/  et ) )
10:8,9,?: e10 28218  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  \/  ch  \/  ta )  <->  ( ps  \/  th  \/  et ) ) ).
qed:10:  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->  ( ( ph  \/  ch  \/  ta )  <->  ( ps  \/  th  \/  et ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th )  /\  ( ta 
 <->  et ) )  ->  ( ( ph  \/  ch 
 \/  ta )  <->  ( ps  \/  th 
 \/  et ) ) )
 
Theoremsbc3orgVD 28389 Virtual deduction proof of sbc3org 28040. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2:1,?: e1_ 28150  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch )  <->  ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch ) ) ).
3::  |-  ( ( ( ph  \/  ps )  \/  ch )  <->  ( ph  \/  ps  \/  ch ) )
32:3:  |-  A. x ( ( ( ph  \/  ps )  \/  ch )  <->  ( ph  \/  ps  \/  ch ) )
33:1,32,?: e10 28218  |-  (. A  e.  B  ->.  [. A  /  x ]. ( ( ( ph  \/  ps )  \/  ch )  <->  ( ph  \/  ps  \/  ch ) ) ).
4:1,33,?: e11 28211  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch )  <->  [. A  /  x ]. ( ph  \/  ps  \/  ch ) ) ).
5:2,4,?: e11 28211  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch ) ) ).
6:1,?: e1_ 28150  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) ).
7:6,?: e1_ 28150  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) ).
8:5,7,?: e11 28211  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) ).
9:?:  |-  ( ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) )
10:8,9,?: e10 28218  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) ) ).
qed:10:  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ].
 ch ) ) )
 
Theorem19.21a3con13vVD 28390* Virtual deduction proof of alrim3con13v 28041. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ph  ->  A. x ph )  ->.  ( ph  ->  A. x ph ) ).
2::  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ( ps  /\  ph  /\  ch ) ).
3:2,?: e2 28154  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ps ).
4:2,?: e2 28154  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ph ).
5:2,?: e2 28154  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ch ).
6:1,4,?: e12 28259  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ph ).
7:3,?: e2 28154  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ps ).
8:5,?: e2 28154  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ch ).
9:7,6,8,?: e222 28159  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ( A. x ps  /\  A. x ph  /\  A. x ch ) ).
10:9,?: e2 28154  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ( ps  /\  ph  /\  ch ) ).
11:10:in2  |-  (. ( ph  ->  A. x ph )  ->.  ( ( ps  /\  ph  /\  ch )  ->  A. x ( ps  /\  ph  /\  ch ) ) ).
qed:11:in1  |-  ( ( ph  ->  A. x ph )  ->  ( ( ps  /\  ph  /\  ch )  ->  A. x ( ps  /\  ph  /\  ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  A. x ph )  ->  ( ( ps  /\  ph  /\  ch )  ->  A. x ( ps 
 /\  ph  /\  ch )
 ) )
 
TheoremexbirVD 28391 Virtual deduction proof of exbir 1365. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->.  ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ).
2::  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ,.  ( ph  /\  ps )  ->.  ( ph  /\  ps ) ).
3::  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ,.  ( ph  /\  ps ) ,  th  ->.  th ).
5:1,2,?: e12 28259  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ,  ( ph  /\  ps )  ->.  ( ch  <->  th ) ).
6:3,5,?: e32 28293  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ,  ( ph  /\  ps ) ,  th  ->.  ch ).
7:6:  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ,  ( ph  /\  ps )  ->.  ( th  ->  ch ) ).
8:7:  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->.  ( ( ph  /\  ps )  ->  ( th  ->  ch ) ) ).
9:8,?: e1_ 28150  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->.  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) ) ).
qed:9:  |-  ( ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) ) )
(Contributed by Alan Sare, 13-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->  ( ph  ->  ( ps  ->  ( th  ->  ch )
 ) ) )
 
TheoremexbiriVD 28392 Virtual deduction proof of exbiri 605. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
h1::  |-  ( ( ph  /\  ps )  ->  ( ch  <->  th ) )
2::  |-  (. ph  ->.  ph ).
3::  |-  (. ph ,. ps  ->.  ps ).
4::  |-  (. ph ,. ps ,. th  ->.  th ).
5:2,1,?: e10 28218  |-  (. ph  ->.  ( ps  ->  ( ch  <->  th ) ) ).
6:3,5,?: e21 28265  |-  (. ph ,. ps  ->.  ( ch  <->  th ) ).
7:4,6,?: e32 28293  |-  (. ph ,. ps ,. th  ->.  ch ).
8:7:  |-  (. ph ,. ps  ->.  ( th  ->  ch ) ).
9:8:  |-  (. ph  ->.  ( ps  ->  ( th  ->  ch ) ) ).
qed:9:  |-  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps )  ->  ( ch  <->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) )
 
Theoremrspsbc2VD 28393* Virtual deduction proof of rspsbc2 28042. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2::  |-  (. A  e.  B ,. C  e.  D  ->.  C  e.  D ).
3::  |-  (. A  e.  B ,. C  e.  D ,. A. x  e.  B  A. y  e.  D ph  ->.  A. x  e.  B A. y  e.  D ph ).
4:1,3,?: e13 28283  |-  (. A  e.  B ,. C  e.  D ,. A. x  e.  B  A. y  e.  D ph  ->.  [. A  /  x ]. A. y  e.  D ph ).
5:1,4,?: e13 28283  |-  (. A  e.  B ,. C  e.  D ,. A. x  e.  B  A. y  e.  D ph  ->.  A. y  e.  D [. A  /  x ]. ph ).
6:2,5,?: e23 28290  |-  (. A  e.  B ,. C  e.  D ,. A. x  e.  B  A. y  e.  D ph  ->.  [. C  /  y ]. [. A  /  x ]. ph ).
7:6:  |-  (. A  e.  B ,. C  e.  D  ->.  ( A. x  e.  B  A. y  e.  D ph  ->  [. C  /  y ]. [. A  /  x ]. ph ) ).
8:7:  |-  (. A  e.  B  ->.  ( C  e.  D  ->  ( A. x  e.  B A. y  e.  D ph  ->  [. C  /  y ]. [. A  /  x ]. ph ) ) ).
qed:8:  |-  ( A  e.  B  ->  ( C  e.  D  ->  ( A. x  e.  B A. y  e.  D ph  ->  [. C  /  y ]. [. A  /  x ]. ph ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  ( C  e.  D  ->  (
 A. x  e.  B  A. y  e.  D  ph  -> 
 [. C  /  y ]. [. A  /  x ].
 ph ) ) )
 
Theorem3impexpVD 28394 Virtual deduction proof of 3impexp 1366. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ph  /\  ps  /\  ch )  ->  th ) ).
2::  |-  ( ( ph  /\  ps  /\  ch )  <->  ( ( ph  /\  ps )  /\  ch ) )
3:1,2,?: e10 28218  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ( ph  /\  ps )  /\  ch )  ->  th ) ).
4:3,?: e1_ 28150  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ph  /\  ps )  ->  ( ch  ->  th ) ) ).
5:4,?: e1_ 28150  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) ).
6:5:  |-  ( ( ( ph  /\  ps  /\  ch )  ->  th )  ->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
7::  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) ).
8:7,?: e1_ 28150  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ph  /\  ps )  ->  ( ch  ->  th ) ) ).
9:8,?: e1_ 28150  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ( ph  /\  ps )  /\  ch )  ->  th ) ).
10:2,9,?: e01 28214  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  th ) ).
11:10:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->  ( ( ph  /\  ps  /\  ch )  ->  th ) )
qed:6,11,?: e00 28303  |-  ( ( ( ph  /\  ps  /\  ch )  ->  th )  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  /\  ps  /\ 
 ch )  ->  th )  <->  (
 ph  ->  ( ps  ->  ( ch  ->  th )
 ) ) )
 
Theorem3impexpbicomVD 28395 Virtual deduction proof of 3impexpbicom 1367. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) ) ).
2::  |-  ( ( th  <->  ta )  <->  ( ta  <->  th ) )
3:1,2,?: e10 28218  |-  (. ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  ( ta  <->  th ) ) ).
4:3,?: e1_ 28150  |-  (. ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )  ->.  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) ).
5:4:  |-  ( ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) )
6::  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )  ->.  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) ).
7:6,?: e1_ 28150  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  ( ta  <->  th ) ) ).
8:7,2,?: e10 28218  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) ) ).
9:8:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )  ->  ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) ) )
qed:5,9,?: e00 28303  |-  ( ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  /\  ps  /\ 
 ch )  ->  ( th 
 <->  ta ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) )
 
Theorem3impexpbicomiVD 28396 Virtual deduction proof of 3impexpbicomi 1368. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
h1::  |-  ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )
qed:1,?: e0_ 28307  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )
 
TheoremsbcoreleleqVD 28397* Virtual deduction proof of sbcoreleleq 28043. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2:1,?: e1_ 28150  |-  (. A  e.  B  ->.  ( [. A  /  y ]. x  e.  y  <->  x  e.  A ) ).
3:1,?: e1_ 28150  |-  (. A  e.  B  ->.  ( [. A  /  y ]. y  e.  x  <->  A  e.  x ) ).
4:1,?: e1_ 28150  |-  (. A  e.  B  ->.  ( [. A  /  y ]. x  =  y  <->  x  =  A ) ).
5:2,3,4,?: e111 28197  |-  (. A  e.  B  ->.  ( ( x  e.  A  \/  A  e.  x  \/  x  =  A )  <->  ( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y ) ) ).
6:1,?: e1_ 28150  |-  (. A  e.  B  ->.  ( [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y )  <->  ( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y ) ) ).
7:5,6: e11 28211  |-  (. A  e.  B  ->.  ( [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y )  <->  ( x  e.  A  \/  A  e.  x  \/  x  =  A ) ) ).
qed:7:  |-  ( A  e.  B  ->  ( [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y )  <->  ( x  e.  A  \/  A  e.  x  \/  x  =  A ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y
 ) 
 <->  ( x  e.  A  \/  A  e.  x  \/  x  =  A )
 ) )
 
Theoremhbra2VD 28398* Virtual deduction proof of nfra2 2673. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  ( A. y  e.  B A. x  e.  A ph  ->  A. y A. y  e.  B A. x  e.  A ph )
2::  |-  ( A. x  e.  A A. y  e.  B ph  <->  A. y  e.  B A. x  e.  A ph )
3:1,2,?: e00 28303  |-  ( A. x  e.  A A. y  e.  B ph  ->  A. y A. y  e.  B A. x  e.  A ph )
4:2:  |-  A. y ( A. x  e.  A A. y  e.  B ph  <->  A. y  e.  B A. x  e.  A ph )
5:4,?: e0_ 28307  |-  ( A. y A. x  e.  A A. y  e.  B ph  <->  A. y A. y  e.  B A. x  e.  A ph )
qed:3,5,?: e00 28303  |-  ( A. x  e.  A A. y  e.  B ph  ->  A. y A. x  e.  A A. y  e.  B ph )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  e.  A  A. y  e.  B  ph  ->  A. y A. x  e.  A  A. y  e.  B  ph )
 
TheoremtratrbVD 28399* Virtual deduction proof of tratrb 28044. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  ( Tr  A  /\  A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ).
2:1,?: e1_ 28150  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  Tr  A ).
3:1,?: e1_ 28150  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) ).
4:1,?: e1_ 28150  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  B  e.  A ).
5::  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  ( x  e.  y  /\  y  e.  B ) ).
6:5,?: e2 28154  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  x  e.  y ).
7:5,?: e2 28154  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  y  e.  B ).
8:2,7,4,?: e121 28179  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  y  e.  A ).
9:2,6,8,?: e122 28176  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  x  e.  A ).
10::  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B ) ,  B  e.  x  ->.  B  e.  x ).
11:6,7,10,?: e223 28158  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B ) ,  B  e.  x  ->.  ( x  e.  y  /\  y  e.  B  /\  B  e.  x ) ).
12:11:  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  ( B  e.  x  ->  ( x  e.  y  /\  y  e.  B  /\  B  e.  x ) ) ).
13::  |-  -.  ( x  e.  y  /\  y  e.  B  /\  B  e.  x )
14:12,13,?: e20 28262  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  -.  B  e.  x ).
15::  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B ) ,  x  =  B  ->.  x  =  B ).
16:7,15,?: e23 28290  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B ) ,  x  =  B  ->.  y  e.  x ).
17:6,16,?: e23 28290  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B ) ,  x  =  B  ->.  ( x  e.  y  /\  y  e.  x ) ).
18:17:  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  ( x  =  B  ->  ( x  e.  y  /\  y  e.  x ) ) ).
19::  |-  -.  ( x  e.  y  /\  y  e.  x )
20:18,19,?: e20 28262  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  -.  x  =  B ).
21:3,?: e1_ 28150  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  A. y  e.  A  A. x  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) ).
22:21,9,4,?: e121 28179  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  [. x  /  x ]. [. B  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y ) ).
23:22,?: e2 28154  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  [. B  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y ) ).
24:4,23,?: e12 28259  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  ( x  e.  B  \/  B  e.  x  \/  x  =  B ) ).
25:14,20,24,?: e222 28159  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  x  e.  B ).
26:25:  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  ( ( x  e.  y  /\  y  e.  B )  ->  x  e.  B ) ).
27::  |-  ( A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  ->  A. y A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
28:27,?: e0_ 28307  |-  ( ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->  A. y ( Tr  A  /\  A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) )
29:28,26:  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  A. y ( ( x  e.  y  /\  y  e.  B )  ->  x  e.  B ) ).
30::  |-  ( A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  ->  A. x A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
31:30,?: e0_ 28307  |-  ( ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->  A. x ( Tr  A  /\  A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) )
32:31,29:  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  A. x  A. y ( ( x  e.  y  /\  y  e.  B )  ->  x  e.  B ) ).
33:32,?: e1_ 28150  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  Tr  B ).
qed:33:  |-  ( ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->  Tr  B )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( Tr  A  /\  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) 
 ->  Tr  B )
 
Theorem3ax5VD 28400 Virtual deduction proof of 3ax5 28045. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. A. x ( ph  ->  ( ps  ->  ch ) )  ->.  A. x ( ph  ->  ( ps  ->  ch ) ) ).
2:1,?: e1_ 28150  |-  (. A. x ( ph  ->  ( ps  ->  ch ) )  ->.  ( A. x ph  ->  A. x ( ps  ->  ch ) ) ).
3::  |-  ( A. x ( ps  ->  ch )  ->  ( A. x ps  ->  A. x ch ) )
4:2,3,?: e10 28218  |-  (. A. x ( ph  ->  ( ps  ->  ch ) )  ->.  ( A. x ph  ->  ( A. x ps  ->  A. x ch ) ) ).
qed:4:  |-  ( A. x ( ph  ->  ( ps  ->  ch ) )  ->  ( A. x ph  ->  ( A. x ps  ->  A. x ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x ( ph  ->  ( ps  ->  ch )
 )  ->  ( A. x ph  ->  ( A. x ps  ->  A. x ch ) ) )
    < 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 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32223
  Copyright terms: Public domain < Previous  Next >