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Theorem List for Metamath Proof Explorer - 28701-28800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremvd12 28701 A virtual deduction with 1 virtual hypothesis virtually inferring a virtual conclusion infers that the same conclusion is virtually inferred by the same virtual hypothesis and an additional hypothesis. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremvd13 28702 A virtual deduction with 1 virtual hypothesis virtually inferring a virtual conclusion infers that the same conclusion is virtually inferred by the same virtual hypothesis and a two additional hypotheses. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremvd23 28703 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.)

Theoremdfvd2imp 28704 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.)

Theoremdfvd2impr 28705 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.)

Theoremin2 28706 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.)

Theoremint2 28707 The virtual deduction introduction rule of converting the end virtual hypothesis of 2 virtual hypotheses into an antecedent. Conventional form of int2 28707 is ex 424. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremiin2 28708 in2 28706 without virtual deductions. (Contributed by Alan Sare, 20-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremin2an 28709 The virtual deduction introduction rule converting the second conjunct of the second virtual hypothesis into the antecedent of the conclusion. exp3a 426 is the non-virtual deduction form of in2an 28709. (Contributed by Alan Sare, 30-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremin3 28710 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.)

Theoremiin3 28711 in3 28710 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremin3an 28712 The virtual deduction introduction rule converting the second conjunct of the third virtual hypothesis into the antecedent of the conclusion. exp4a 590 is the non-virtual deduction form of in3an 28712. (Contributed by Alan Sare, 25-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremint3 28713 The virtual deduction introduction rule of converting the end virtual hypothesis of 3 virtual hypotheses into an antecedent. Conventional form of int3 28713 is 3expia 1155. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremidn2 28714 Virtual deduction identity rule which is idd 22 with virtual deduction symbols. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremiden2 28715 Virtual deduction identity rule. simpr 448 in conjunction form Virtual Deduction notation. (Contributed by Alan Sare, 5-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremidn3 28716 Virtual deduction identity rule for 3 virtual hypotheses. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremgen11 28717* Virtual deduction generalizing rule for 1 quantifying variable and 1 virtual hypothesis. alrimiv 1641 is gen11 28717 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremgen11nv 28718 Virtual deduction generalizing rule for 1 quantifying variable and 1 virtual hypothesis without distinct variables. alrimih 1574 is gen11nv 28718 without virtual deductions. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremgen12 28719* Virtual deduction generalizing rule for 2 quantifying variables and 1 virtual hypothesis. gen12 28719 is alrimivv 1642 with virtual deductions. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremgen21 28720* Virtual deduction generalizing rule for 1 quantifying variables and 2 virtual hypothesis. gen21 28720 is alrimdv 1643 with virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremgen21nv 28721 Virtual deduction form of alrimdh 1597. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremgen31 28722* Virtual deduction generalizing rule for 1 quantifying variable and 3 virtual hypothesis. gen31 28722 is ggen31 28631 with virtual deductions. (Contributed by Alan Sare, 22-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremgen22 28723* 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.)

Theoremggen22 28724* gen22 28723 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremexinst 28725 Existential Instantiation. Virtual deduction form of exlimexi 28608. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremexinst01 28726 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 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.)

Theoremexinst11 28727 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 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.)

Theoreme1_ 28728 A Virtual deduction elimination rule. syl 16 is e1_ 28728 without virtual deductions. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremel1 28729 A Virtual deduction elimination rule. syl 16 is el1 28729 without virtual deductions. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme1bi 28730 Biconditional form of e1_ 28728. sylib 189 is e1bi 28730 without virtual deductions. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme1bir 28731 Right biconditional form of e1_ 28728. sylibr 204 is e1bir 28731 without virtual deductions. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme2 28732 A virtual deduction elimination rule. syl6 31 is e2 28732 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme2bi 28733 Bi-conditional form of e2 28732. syl6ib 218 is e2bi 28733 without virtual deductions. (Contributed by Alan Sare, 10-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme2bir 28734 Right bi-conditional form of e2 28732. syl6ibr 219 is e2bir 28734 without virtual deductions. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee223 28735 e223 28736 without virtual deductions. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme223 28736 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme222 28737 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme220 28738 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee220 28739 e220 28738 without virtual deductions. (Contributed by Alan Sare, 12-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme202 28740 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee202 28741 e202 28740 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme022 28742 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee022 28743 e022 28742 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme002 28744 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee002 28745 e002 28744 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme020 28746 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee020 28747 e020 28746 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme200 28748 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee200 28749 e200 28748 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme221 28750 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee221 28751 e221 28750 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme212 28752 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee212 28753 e212 28752 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme122 28754 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme112 28755 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee112 28756 e112 28755 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme121 28757 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme211 28758 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee211 28759 e211 28758 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme210 28760 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee210 28761 e210 28760 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme201 28762 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee201 28763 e201 28762 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme120 28764 A virtual deduction elimination rule. (Contributed by Alan Sare, 10-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee120 28765 Virtual deduction rule e120 28764 without virtual deduction symbols. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme021 28766 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee021 28767 e021 28766 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme012 28768 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee012 28769 e012 28768 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme102 28770 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee102 28771 e102 28770 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme22 28772 A virtual deduction elimination rule. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme22an 28773 Conjunction form of e22 28772. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee22an 28774 e22an 28773 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme111 28775 A virtual deduction elimination rule. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme1111 28776 A virtual deduction elimination rule. (Contributed by Alan Sare, 6-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme110 28777 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee110 28778 e110 28777 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme101 28779 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee101 28780 e101 28779 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme011 28781 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee011 28782 e011 28781 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme100 28783 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee100 28784 e100 28783 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme010 28785 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee010 28786 e010 28785 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme001 28787 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee001 28788 e001 28787 without virtual deductions. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme11 28789 A virtual deduction elimination rule. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme11an 28790 Conjunction form of e11 28789. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee11an 28791 e11an 28790 without virtual deductions. syl22anc 1185 is also e11an 28790 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.)

Theoreme01 28792 A virtual deduction elimination rule. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme01an 28793 Conjunction form of e01 28792. (Contributed by Alan Sare, 11-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee01an 28794 e01an 28793 without virtual deductions. sylancr 645 is also a form of e01an 28793 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.)

Theoreme10 28795 A virtual deduction elimination rule. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme10an 28796 Conjunction form of e10 28795. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee10an 28797 e10an 28796 without virtual deductions. sylancl 644 is also e10an 28796 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.)

Theoreme02 28798 A virtual deduction elimination rule. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoreme02an 28799 Conjunction form of e02 28798. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee02an 28800 e02an 28799 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

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