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Theorem List for Metamath Proof Explorer - 33801-33900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremvd23 33801 A virtual deduction with 2 virtual hypotheses virtually inferring a virtual conclusion infers that the same conclusion is virtually inferred by the same 2 virtual hypotheses and a third hypothesis. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   =>    |- 
 (. ph ,. ps ,. th  ->.  ch ).
 
Theoremdfvd2imp 33802 The virtual deduction form of a 2-antecedent nested implication implies the 2-antecedent nested implication. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( (. ph ,. ps  ->.  ch ).  ->  (
 ph  ->  ( ps  ->  ch ) ) )
 
Theoremdfvd2impr 33803 A 2-antecedent nested implication implies its virtual deduction form. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  (. ph ,. ps  ->.  ch ). )
 
Theoremin2 33804 The virtual deduction introduction rule of converting the end virtual hypothesis of 2 virtual hypotheses into an antecedent. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   =>    |- 
 (. ph  ->.  ( ps  ->  ch ) ).
 
Theoremint2 33805 The virtual deduction introduction rule of converting the end virtual hypothesis of 2 virtual hypotheses into an antecedent. Conventional form of int2 33805 is ex 432. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (. ph
 ,. ps ).  ->.  ch ).   =>    |-  (. ph  ->.  ( ps  ->  ch ) ).
 
Theoremiin2 33806 in2 33804 without virtual deductions. (Contributed by Alan Sare, 20-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  ch ) )
 
Theoremin2an 33807 The virtual deduction introduction rule converting the second conjunct of the second virtual hypothesis into the antecedent of the conclusion. expd 434 is the non-virtual deduction form of in2an 33807. (Contributed by Alan Sare, 30-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ( ps  /\  ch ) 
 ->.  th ).   =>    |- 
 (. ph ,. ps  ->.  ( ch  ->  th ) ).
 
Theoremin3 33808 The virtual deduction introduction rule of converting the end virtual hypothesis of 3 virtual hypotheses into an antecedent. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ch  ->.  th ).   =>    |-  (. ph ,. ps  ->.  ( ch  ->  th ) ).
 
Theoremiin3 33809 in3 33808 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
 
Theoremin3an 33810 The virtual deduction introduction rule converting the second conjunct of the third virtual hypothesis into the antecedent of the conclusion. exp4a 604 is the non-virtual deduction form of in3an 33810. (Contributed by Alan Sare, 25-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ( ch 
 /\  th )  ->.  ta ).   =>    |-  (. ph ,. ps ,. ch  ->.  ( th  ->  ta ) ).
 
Theoremint3 33811 The virtual deduction introduction rule of converting the end virtual hypothesis of 3 virtual hypotheses into an antecedent. Conventional form of int3 33811 is 3expia 1196. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (. ph
 ,. ps ,. ch ).  ->.  th
 ).   =>    |- 
 (. (. ph ,. ps ).  ->.  ( ch  ->  th ) ).
 
Theoremidn2 33812 Virtual deduction identity rule which is idd 24 with virtual deduction symbols. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ps ).
 
Theoremiden2 33813 Virtual deduction identity rule. simpr 459 in conjunction form Virtual Deduction notation. (Contributed by Alan Sare, 5-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. (. ph
 ,. ps ).  ->.  ps ).
 
Theoremidn3 33814 Virtual deduction identity rule for 3 virtual hypotheses. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ch  ->.  ch ).
 
Theoremgen11 33815* Virtual deduction generalizing rule for 1 quantifying variable and 1 virtual hypothesis. alrimiv 1724 is gen11 33815 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   =>    |- 
 (. ph  ->.  A. x ps ).
 
Theoremgen11nv 33816 Virtual deduction generalizing rule for 1 quantifying variable and 1 virtual hypothesis without distinct variables. alrimih 1647 is gen11nv 33816 without virtual deductions. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  (. ph  ->.  ps
 ).   =>    |- 
 (. ph  ->.  A. x ps ).
 
Theoremgen12 33817* Virtual deduction generalizing rule for 2 quantifying variables and 1 virtual hypothesis. gen12 33817 is alrimivv 1725 with virtual deductions. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   =>    |- 
 (. ph  ->.  A. x A. y ps ).
 
Theoremgen21 33818* Virtual deduction generalizing rule for 1 quantifying variables and 2 virtual hypothesis. gen21 33818 is alrimdv 1726 with virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   =>    |- 
 (. ph ,. ps  ->.  A. x ch ).
 
Theoremgen21nv 33819 Virtual deduction form of alrimdh 1677. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. x ps )   &    |-  (. ph ,. ps  ->.  ch ).   =>    |- 
 (. ph ,. ps  ->.  A. x ch ).
 
Theoremgen31 33820* Virtual deduction generalizing rule for 1 quantifying variable and 3 virtual hypothesis. gen31 33820 is ggen31 33730 with virtual deductions. (Contributed by Alan Sare, 22-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps ,. ch  ->.  th ).   =>    |-  (. ph ,. ps ,. ch  ->.  A. x th ).
 
Theoremgen22 33821* Virtual deduction generalizing rule for 2 quantifying variables and 2 virtual hypothesis. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   =>    |- 
 (. ph ,. ps  ->.  A. x A. y ch ).
 
Theoremggen22 33822* gen22 33821 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x A. y ch ) )
 
Theoremexinst 33823 Existential Instantiation. Virtual deduction form of exlimexi 33700. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ps  ->  A. x ps )   &    |-  (. E. x ph ,. ph  ->.  ps ).   =>    |-  ( E. x ph 
 ->  ps )
 
Theoremexinst01 33824 Existential Instantiation. Virtual Deduction rule corresponding to a special case of the Natural Deduction Sequent Calculus rule called Rule C in [Margaris] p. 79 and E  E. in Table 1 on page 4 of the paper "Extracting information from intermediate T-systems" (2000) presented at IMLA99 by Mauro Ferrari, Camillo Fiorentini, and Pierangelo Miglioli. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  E. x ps   &    |- 
 (. ph ,. ps  ->.  ch ).   &    |-  ( ph  ->  A. x ph )   &    |-  ( ch  ->  A. x ch )   =>    |-  (. ph  ->.  ch
 ).
 
Theoremexinst11 33825 Existential Instantiation. Virtual Deduction rule corresponding to a special case of the Natural Deduction Sequent Calculus rule called Rule C in [Margaris] p. 79 and E  E. in Table 1 on page 4 of the paper "Extracting information from intermediate T-systems" (2000) presented at IMLA99 by Mauro Ferrari, Camillo Fiorentini, and Pierangelo Miglioli. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  E. x ps ).   &    |-  (. ph ,. ps  ->.  ch ).   &    |-  ( ph  ->  A. x ph )   &    |-  ( ch  ->  A. x ch )   =>    |-  (. ph  ->.  ch
 ).
 
Theoreme1a 33826 A Virtual deduction elimination rule. syl 16 is e1a 33826 without virtual deductions. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |-  ( ps  ->  ch )   =>    |-  (. ph  ->.  ch
 ).
 
Theoremel1 33827 A Virtual deduction elimination rule. syl 16 is el1 33827 without virtual deductions. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |-  ( ps  ->  ch )   =>    |-  (. ph  ->.  ch
 ).
 
Theoreme1bi 33828 Biconditional form of e1a 33826. sylib 196 is e1bi 33828 without virtual deductions. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |-  ( ps  <->  ch )   =>    |- 
 (. ph  ->.  ch ).
 
Theoreme1bir 33829 Right biconditional form of e1a 33826. sylibr 212 is e1bir 33829 without virtual deductions. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |-  ( ch  <->  ps )   =>    |- 
 (. ph  ->.  ch ).
 
Theoreme2 33830 A virtual deduction elimination rule. syl6 33 is e2 33830 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  ( ch  ->  th )   =>    |- 
 (. ph ,. ps  ->.  th ).
 
Theoreme2bi 33831 Biconditional form of e2 33830. syl6ib 226 is e2bi 33831 without virtual deductions. (Contributed by Alan Sare, 10-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  ( ch  <->  th )   =>    |- 
 (. ph ,. ps  ->.  th ).
 
Theoreme2bir 33832 Right biconditional form of e2 33830. syl6ibr 227 is e2bir 33832 without virtual deductions. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  ( th  <->  ch )   =>    |- 
 (. ph ,. ps  ->.  th ).
 
Theoremee223 33833 e223 33834 without virtual deductions. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th )
 )   &    |-  ( ph  ->  ( ps  ->  ( ta  ->  et ) ) )   &    |-  ( ch  ->  ( th  ->  ( et  ->  ze )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ta  ->  ze ) ) )
 
Theoreme223 33834 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph ,. ps  ->. 
 th ).   &    |-  (. ph ,. ps ,. ta  ->.  et ).   &    |-  ( ch  ->  ( th  ->  ( et  ->  ze )
 ) )   =>    |- 
 (. ph ,. ps ,. ta  ->.  ze ).
 
Theoreme222 33835 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 ).   &    |-  (. ph ,. ps  ->.  ta ).   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. ps  ->.  et ).
 
Theoreme220 33836 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  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. ps  ->.  et ).
 
Theoremee220 33837 e220 33836 without virtual deductions. (Contributed by Alan Sare, 12-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th )
 )   &    |- 
 ta   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et )
 )
 
Theoreme202 33838 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  th   &    |-  (. ph ,. ps  ->.  ta ).   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. ps  ->.  et ).
 
Theoremee202 33839 e202 33838 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  th   &    |-  ( ph  ->  ( ps  ->  ta )
 )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et )
 )
 
Theoreme022 33840 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps ,. ch  ->. 
 th ).   &    |-  (. ps ,. ch  ->.  ta ).   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ps ,. ch  ->.  et ).
 
Theoremee022 33841 e022 33840 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ( ch  ->  th )
 )   &    |-  ( ps  ->  ( ch  ->  ta ) )   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ps  ->  ( ch  ->  et ) )
 
Theoreme002 33842 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  (. ch ,. th  ->.  ta ).   &    |-  ( ph  ->  ( ps  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ch ,. th  ->.  et ).
 
Theoremee002 33843 e002 33842 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  ( ch  ->  ( th  ->  ta )
 )   &    |-  ( ph  ->  ( ps  ->  ( ta  ->  et ) ) )   =>    |-  ( ch  ->  ( th  ->  et )
 )
 
Theoreme020 33844 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps ,. ch  ->. 
 th ).   &    |-  ta   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ps ,. ch  ->.  et ).
 
Theoremee020 33845 e020 33844 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ( ch  ->  th )
 )   &    |- 
 ta   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ps  ->  ( ch  ->  et )
 )
 
Theoreme200 33846 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  th   &    |-  ta   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. ps  ->.  et ).
 
Theoremee200 33847 e200 33846 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  th   &    |-  ta   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoreme221 33848 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph ,. ps  ->. 
 th ).   &    |-  (. ph  ->.  ta ).   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ph ,. ps  ->.  et ).
 
Theoremee221 33849 e221 33848 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th )
 )   &    |-  ( ph  ->  ta )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoreme212 33850 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph  ->.  th ).   &    |-  (. ph ,. ps  ->.  ta ).   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. ps  ->.  et ).
 
Theoremee212 33851 e212 33850 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ( ps  ->  ta )
 )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et )
 )
 
Theoreme122 33852 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph ,. ch  ->.  th ).   &    |-  (. ph ,. ch  ->.  ta ).   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. ch  ->.  et ).
 
Theoreme112 33853 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph  ->.  ch ).   &    |-  (. ph ,. th  ->.  ta ).   &    |-  ( ps  ->  ( ch  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ph ,. th  ->.  et ).
 
Theoremee112 33854 e112 33853 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( th  ->  ta )
 )   &    |-  ( ps  ->  ( ch  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( th  ->  et )
 )
 
Theoreme121 33855 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph ,. ch  ->.  th ).   &    |-  (. ph  ->.  ta
 ).   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (. ph ,. ch  ->.  et ).
 
Theoreme211 33856 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph  ->.  th ).   &    |-  (. ph  ->.  ta
 ).   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  (. ph ,. ps  ->.  et ).
 
Theoremee211 33857 e211 33856 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoreme210 33858 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph  ->.  th ).   &    |-  ta   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ph ,. ps  ->.  et ).
 
Theoremee210 33859 e210 33858 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  th )   &    |-  ta   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoreme201 33860 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  th   &    |-  (. ph  ->.  ta ).   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ph ,. ps  ->.  et ).
 
Theoremee201 33861 e201 33860 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  th   &    |-  ( ph  ->  ta )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoreme120 33862 A virtual deduction elimination rule. (Contributed by Alan Sare, 10-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph ,. ch  ->.  th ).   &    |-  ta   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ph ,. ch  ->.  et ).
 
Theoremee120 33863 Virtual deduction rule e120 33862 without virtual deduction symbols. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   &    |- 
 ta   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( ch  ->  et )
 )
 
Theoreme021 33864 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps ,. ch  ->. 
 th ).   &    |-  (. ps  ->.  ta ).   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ps ,. ch  ->.  et ).
 
Theoremee021 33865 e021 33864 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ( ch  ->  th )
 )   &    |-  ( ps  ->  ta )   &    |-  ( ph  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ps  ->  ( ch  ->  et ) )
 
Theoreme012 33866 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps  ->.  ch ).   &    |-  (. ps ,. th  ->.  ta ).   &    |-  ( ph  ->  ( ch  ->  ( ta  ->  et ) ) )   =>    |-  (.
 ps ,. th  ->.  et ).
 
Theoremee012 33867 e012 33866 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  ( ps  ->  ( th  ->  ta )
 )   &    |-  ( ph  ->  ( ch  ->  ( ta  ->  et ) ) )   =>    |-  ( ps  ->  ( th  ->  et )
 )
 
Theoreme102 33868 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 ch   &    |- 
 (. ph ,. th  ->.  ta ).   &    |-  ( ps  ->  ( ch  ->  ( ta  ->  et )
 ) )   =>    |- 
 (. ph ,. th  ->.  et ).
 
Theoremee102 33869 e102 33868 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  ( ph  ->  ( th  ->  ta )
 )   &    |-  ( ps  ->  ( ch  ->  ( ta  ->  et ) ) )   =>    |-  ( ph  ->  ( th  ->  et )
 )
 
Theoreme22 33870 A virtual deduction elimination rule. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph ,. ps  ->. 
 th ).   &    |-  ( ch  ->  ( th  ->  ta )
 )   =>    |- 
 (. ph ,. ps  ->.  ta ).
 
Theoreme22an 33871 Conjunction form of e22 33870. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph ,. ps  ->.  ch ).   &    |-  (. ph ,. ps  ->. 
 th ).   &    |-  ( ( ch 
 /\  th )  ->  ta )   =>    |-  (. ph ,. ps  ->.  ta ).
 
Theoremee22an 33872 e22an 33871 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th )
 )   &    |-  ( ( ch  /\  th )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ta ) )
 
Theoreme111 33873 A virtual deduction elimination rule (see syl3c 61). (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph  ->.  ch ).   &    |-  (. ph  ->.  th ).   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) )   =>    |- 
 (. ph  ->.  ta ).
 
Theoreme1111 33874 A virtual deduction elimination rule. (Contributed by Alan Sare, 6-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph  ->.  ch ).   &    |-  (. ph  ->.  th ).   &    |-  (. ph  ->.  ta
 ).   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) )   =>    |-  (. ph  ->.  et ).
 
Theoreme110 33875 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph  ->.  ch ).   &    |-  th   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  (.
 ph 
 ->.  ta ).
 
Theoremee110 33876 e110 33875 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  th   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  ( ph  ->  ta )
 
Theoreme101 33877 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 ch   &    |- 
 (. ph  ->.  th ).   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  (.
 ph 
 ->.  ta ).
 
Theoremee101 33878 e101 33877 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  ( ph  ->  th )   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  ( ph  ->  ta )
 
Theoreme011 33879 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps  ->.  ch ).   &    |-  (. ps  ->.  th
 ).   &    |-  ( ph  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  (. ps  ->.  ta ).
 
Theoremee011 33880 e011 33879 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  ( ps  ->  th )   &    |-  ( ph  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  ( ps  ->  ta )
 
Theoreme100 33881 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 ch   &    |- 
 th   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  (. ph  ->.  ta ).
 
Theoremee100 33882 e100 33881 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  th   &    |-  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  ( ph  ->  ta )
 
Theoreme010 33883 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps  ->.  ch ).   &    |-  th   &    |-  ( ph  ->  ( ch  ->  ( th  ->  ta )
 ) )   =>    |- 
 (. ps  ->.  ta ).
 
Theoremee010 33884 e010 33883 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  th   &    |-  ( ph  ->  ( ch  ->  ( th  ->  ta ) ) )   =>    |-  ( ps  ->  ta )
 
Theoreme001 33885 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  (. ch  ->.  th ).   &    |-  ( ph  ->  ( ps  ->  ( th  ->  ta )
 ) )   =>    |- 
 (. ch  ->.  ta ).
 
Theoremee001 33886 e001 33885 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ps   &    |-  ( ch  ->  th )   &    |-  ( ph  ->  ( ps  ->  ( th  ->  ta ) ) )   =>    |-  ( ch  ->  ta )
 
Theoreme11 33887 A virtual deduction elimination rule. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph  ->.  ch ).   &    |-  ( ps  ->  ( ch  ->  th )
 )   =>    |- 
 (. ph  ->.  th ).
 
Theoreme11an 33888 Conjunction form of e11 33887. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 (. ph  ->.  ch ).   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (. ph  ->.  th
 ).
 
Theoremee11an 33889 e11an 33888 without virtual deductions. syl22anc 1227 is also e11an 33888 without virtual deductions, exept with a different order of hypotheses. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoreme01 33890 A virtual deduction elimination rule. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps  ->.  ch ).   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |- 
 (. ps  ->.  th ).
 
Theoreme01an 33891 Conjunction form of e01 33890. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps  ->.  ch ).   &    |-  (
 ( ph  /\  ch )  ->  th )   =>    |- 
 (. ps  ->.  th ).
 
Theoremee01an 33892 e01an 33891 without virtual deductions. sylancr 661 is also a form of e01an 33891 without virtual deduction, except the order of the hypotheses is different. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  ( ( ph  /\ 
 ch )  ->  th )   =>    |-  ( ps  ->  th )
 
Theoreme10 33893 A virtual deduction elimination rule (see mpisyl 18). (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 ch   &    |-  ( ps  ->  ( ch  ->  th ) )   =>    |-  (. ph  ->.  th ).
 
Theoreme10an 33894 Conjunction form of e10 33893. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  ps
 ).   &    |- 
 ch   &    |-  ( ( ps  /\  ch )  ->  th )   =>    |-  (. ph  ->.  th
 ).
 
Theoremee10an 33895 e10an 33894 without virtual deductions. sylancl 660 is also e10an 33894 without virtual deductions, except the order of the hypotheses is different. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoreme02 33896 A virtual deduction elimination rule. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps ,. ch  ->. 
 th ).   &    |-  ( ph  ->  ( th  ->  ta )
 )   =>    |- 
 (. ps ,. ch  ->.  ta ).
 
Theoreme02an 33897 Conjunction form of e02 33896. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. ps ,. ch  ->. 
 th ).   &    |-  ( ( ph  /\ 
 th )  ->  ta )   =>    |-  (. ps ,. ch  ->.  ta ).
 
Theoremee02an 33898 e02an 33897 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ( ch  ->  th )
 )   &    |-  ( ( ph  /\  th )  ->  ta )   =>    |-  ( ps  ->  ( ch  ->  ta ) )
 
Theoremeel021old 33899 el021old 33900 without virtual deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ( ps 
 /\  ch )  ->  th )   &    |-  (
 ( ph  /\  th )  ->  ta )   =>    |-  ( ( ps  /\  ch )  ->  ta )
 
Theoremel021old 33900 A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  (. (. ps ,. ch ).  ->.  th ).   &    |-  (
 ( ph  /\  th )  ->  ta )   =>    |- 
 (. (. ps ,. ch ).  ->.  ta ).
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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 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38521
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