P. 10 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 901-1000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dn1dc 901	DN1 for decidable propositions. Without the decidability conditions, DN1 can serve as a single axiom for Boolean algebra. See http://www-unix.mcs.anl.gov/~mccune/papers/basax/v12.pdf. (Contributed by Jim Kingdon, 22-Apr-2018.)
|- ( (DECID ph /\ (DECID ps /\ (DECID ch /\ DECID th ) ) ) -> ( -. ( -. ( -. ( ph \/ ps ) \/ ch ) \/ -. ( ph \/ -. ( -. ch \/ -. ( ch \/ th ) ) ) ) <-> ch ) )
Theorem	pm5.71dc 902	Decidable proposition version of theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.) (Modified for decidability by Jim Kingdon, 19-Apr-2018.)
|- (DECID ps -> ( ( ps -> -. ch ) -> ( ( ( ph \/ ps ) /\ ch ) <-> ( ph /\ ch ) ) ) )
Theorem	pm5.75 903	Theorem *5.75 of [WhiteheadRussell] p. 126. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 23-Dec-2012.)
|- ( ( ( ch -> -. ps ) /\ ( ph <-> ( ps \/ ch ) ) ) -> ( ( ph /\ -. ps ) <-> ch ) )
Theorem	bimsc1 904	Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ( ( ph -> ps ) /\ ( ch <-> ( ps /\ ph ) ) ) -> ( ch <-> ph ) )
Theorem	ccase 905	Inference for combining cases. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)
|- ( ( ph /\ ps ) -> ta )   &    |- ( ( ch /\ ps ) -> ta )   &    |- ( ( ph /\ th ) -> ta )   &    |- ( ( ch /\ th ) -> ta )   =>    |- ( ( ( ph \/ ch ) /\ ( ps \/ th ) ) -> ta )
Theorem	ccased 906	Deduction for combining cases. (Contributed by NM, 9-May-2004.)
|- ( ph -> ( ( ps /\ ch ) -> et ) )   &    |- ( ph -> ( ( th /\ ch ) -> et ) )   &    |- ( ph -> ( ( ps /\ ta ) -> et ) )   &    |- ( ph -> ( ( th /\ ta ) -> et ) )   =>    |- ( ph -> ( ( ( ps \/ th ) /\ ( ch \/ ta ) ) -> et ) )
Theorem	ccase2 907	Inference for combining cases. (Contributed by NM, 29-Jul-1999.)
|- ( ( ph /\ ps ) -> ta )   &    |- ( ch -> ta )   &    |- ( th -> ta )   =>    |- ( ( ( ph \/ ch ) /\ ( ps \/ th ) ) -> ta )
Theorem	niabn 908	Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
|- ph   =>    |- ( -. ps -> ( ( ch /\ ps ) <-> -. ph ) )
Theorem	dedlem0a 909	Alternate version of dedlema 910. (Contributed by NM, 2-Apr-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
|- ( ph -> ( ps <-> ( ( ch -> ph ) -> ( ps /\ ph ) ) ) )
Theorem	dedlema 910	Lemma for iftrue 3356. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
|- ( ph -> ( ps <-> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) ) )
Theorem	dedlemb 911	Lemma for iffalse 3359. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)
|- ( -. ph -> ( ch <-> ( ( ps /\ ph ) \/ ( ch /\ -. ph ) ) ) )
Theorem	pm4.42r 912	One direction of Theorem *4.42 of [WhiteheadRussell] p. 119. (Contributed by Jim Kingdon, 4-Aug-2018.)
|- ( ( ( ph /\ ps ) \/ ( ph /\ -. ps ) ) -> ph )
Theorem	ninba 913	Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
|- ph   =>    |- ( -. ps -> ( -. ph <-> ( ch /\ ps ) ) )
Theorem	prlem1 914	A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)
|- ( ph -> ( et <-> ch ) )   &    |- ( ps -> -. th )   =>    |- ( ph -> ( ps -> ( ( ( ps /\ ch ) \/ ( th /\ ta ) ) -> et ) ) )
Theorem	prlem2 915	A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
|- ( ( ( ph /\ ps ) \/ ( ch /\ th ) ) <-> ( ( ph \/ ch ) /\ ( ( ph /\ ps ) \/ ( ch /\ th ) ) ) )
Theorem	oplem1 916	A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
|- ( ph -> ( ps \/ ch ) )   &    |- ( ph -> ( th \/ ta ) )   &    |- ( ps <-> th )   &    |- ( ch -> ( th <-> ta ) )   =>    |- ( ph -> ps )
Theorem	rnlem 917	Lemma used in construction of real numbers. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> ( ( ( ph /\ ch ) /\ ( ps /\ th ) ) /\ ( ( ph /\ th ) /\ ( ps /\ ch ) ) ) )
1.2.12  Abbreviated conjunction and disjunction of three wff's
Syntax	w3o 918	Extend wff definition to include 3-way disjunction ('or').
wff ( ph \/ ps \/ ch )
Syntax	w3a 919	Extend wff definition to include 3-way conjunction ('and').
wff ( ph /\ ps /\ ch )
Definition	df-3or 920	Define disjunction ('or') of 3 wff's. Definition *2.33 of [WhiteheadRussell] p. 105. This abbreviation reduces the number of parentheses and emphasizes that the order of bracketing is not important by virtue of the associative law orass 716. (Contributed by NM, 8-Apr-1994.)
|- ( ( ph \/ ps \/ ch ) <-> ( ( ph \/ ps ) \/ ch ) )
Definition	df-3an 921	Define conjunction ('and') of 3 wff.s. Definition *4.34 of [WhiteheadRussell] p. 118. This abbreviation reduces the number of parentheses and emphasizes that the order of bracketing is not important by virtue of the associative law anass 393. (Contributed by NM, 8-Apr-1994.)
|- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
Theorem	3orass 922	Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)
|- ( ( ph \/ ps \/ ch ) <-> ( ph \/ ( ps \/ ch ) ) )
Theorem	3anass 923	Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
|- ( ( ph /\ ps /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) )
Theorem	3anrot 924	Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
|- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ch /\ ph ) )
Theorem	3orrot 925	Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.)
|- ( ( ph \/ ps \/ ch ) <-> ( ps \/ ch \/ ph ) )
Theorem	3ancoma 926	Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
|- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ph /\ ch ) )
Theorem	3ancomb 927	Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
|- ( ( ph /\ ps /\ ch ) <-> ( ph /\ ch /\ ps ) )
Theorem	3orcomb 928	Commutation law for triple disjunction. (Contributed by Scott Fenton, 20-Apr-2011.)
|- ( ( ph \/ ps \/ ch ) <-> ( ph \/ ch \/ ps ) )
Theorem	3anrev 929	Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
|- ( ( ph /\ ps /\ ch ) <-> ( ch /\ ps /\ ph ) )
Theorem	3anan32 930	Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ch ) /\ ps ) )
Theorem	3anan12 931	Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ( ph /\ ch ) ) )
Theorem	anandi3 932	Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
|- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ( ph /\ ch ) ) )
Theorem	anandi3r 933	Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
|- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ( ch /\ ps ) ) )
Theorem	3ioran 934	Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
|- ( -. ( ph \/ ps \/ ch ) <-> ( -. ph /\ -. ps /\ -. ch ) )
Theorem	3simpa 935	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
|- ( ( ph /\ ps /\ ch ) -> ( ph /\ ps ) )
Theorem	3simpb 936	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
|- ( ( ph /\ ps /\ ch ) -> ( ph /\ ch ) )
Theorem	3simpc 937	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( ph /\ ps /\ ch ) -> ( ps /\ ch ) )
Theorem	simp1 938	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
|- ( ( ph /\ ps /\ ch ) -> ph )
Theorem	simp2 939	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
|- ( ( ph /\ ps /\ ch ) -> ps )
Theorem	simp3 940	Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
|- ( ( ph /\ ps /\ ch ) -> ch )
Theorem	simpl1 941	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
|- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ph )
Theorem	simpl2 942	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
|- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ps )
Theorem	simpl3 943	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
|- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ch )
Theorem	simpr1 944	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
|- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ps )
Theorem	simpr2 945	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
|- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ch )
Theorem	simpr3 946	Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
|- ( ( ph /\ ( ps /\ ch /\ th ) ) -> th )
Theorem	simp1i 947	Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
|- ( ph /\ ps /\ ch )   =>    |- ph
Theorem	simp2i 948	Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
|- ( ph /\ ps /\ ch )   =>    |- ps
Theorem	simp3i 949	Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
|- ( ph /\ ps /\ ch )   =>    |- ch
Theorem	simp1d 950	Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
|- ( ph -> ( ps /\ ch /\ th ) )   =>    |- ( ph -> ps )
Theorem	simp2d 951	Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
|- ( ph -> ( ps /\ ch /\ th ) )   =>    |- ( ph -> ch )
Theorem	simp3d 952	Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
|- ( ph -> ( ps /\ ch /\ th ) )   =>    |- ( ph -> th )
Theorem	simp1bi 953	Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph <-> ( ps /\ ch /\ th ) )   =>    |- ( ph -> ps )
Theorem	simp2bi 954	Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph <-> ( ps /\ ch /\ th ) )   =>    |- ( ph -> ch )
Theorem	simp3bi 955	Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph <-> ( ps /\ ch /\ th ) )   =>    |- ( ph -> th )
Theorem	3adant1 956	Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( th /\ ph /\ ps ) -> ch )
Theorem	3adant2 957	Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ph /\ th /\ ps ) -> ch )
Theorem	3adant3 958	Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ph /\ ps /\ th ) -> ch )
Theorem	3ad2ant1 959	Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
|- ( ph -> ch )   =>    |- ( ( ph /\ ps /\ th ) -> ch )
Theorem	3ad2ant2 960	Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
|- ( ph -> ch )   =>    |- ( ( ps /\ ph /\ th ) -> ch )
Theorem	3ad2ant3 961	Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
|- ( ph -> ch )   =>    |- ( ( ps /\ th /\ ph ) -> ch )
Theorem	simp1l 962	Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
|- ( ( ( ph /\ ps ) /\ ch /\ th ) -> ph )
Theorem	simp1r 963	Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
|- ( ( ( ph /\ ps ) /\ ch /\ th ) -> ps )
Theorem	simp2l 964	Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
|- ( ( ph /\ ( ps /\ ch ) /\ th ) -> ps )
Theorem	simp2r 965	Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
|- ( ( ph /\ ( ps /\ ch ) /\ th ) -> ch )
Theorem	simp3l 966	Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
|- ( ( ph /\ ps /\ ( ch /\ th ) ) -> ch )
Theorem	simp3r 967	Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
|- ( ( ph /\ ps /\ ( ch /\ th ) ) -> th )
Theorem	simp11 968	Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
|- ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) -> ph )
Theorem	simp12 969	Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
|- ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) -> ps )
Theorem	simp13 970	Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
|- ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) -> ch )
Theorem	simp21 971	Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
|- ( ( ph /\ ( ps /\ ch /\ th ) /\ ta ) -> ps )
Theorem	simp22 972	Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
|- ( ( ph /\ ( ps /\ ch /\ th ) /\ ta ) -> ch )
Theorem	simp23 973	Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
|- ( ( ph /\ ( ps /\ ch /\ th ) /\ ta ) -> th )
Theorem	simp31 974	Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
|- ( ( ph /\ ps /\ ( ch /\ th /\ ta ) ) -> ch )
Theorem	simp32 975	Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
|- ( ( ph /\ ps /\ ( ch /\ th /\ ta ) ) -> th )
Theorem	simp33 976	Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
|- ( ( ph /\ ps /\ ( ch /\ th /\ ta ) ) -> ta )
Theorem	simpll1 977	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta ) -> ph )
Theorem	simpll2 978	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta ) -> ps )
Theorem	simpll3 979	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta ) -> ch )
Theorem	simplr1 980	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta ) -> ph )
Theorem	simplr2 981	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta ) -> ps )
Theorem	simplr3 982	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta ) -> ch )
Theorem	simprl1 983	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ph )
Theorem	simprl2 984	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ps )
Theorem	simprl3 985	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ch )
Theorem	simprr1 986	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ph )
Theorem	simprr2 987	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ps )
Theorem	simprr3 988	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ch )
Theorem	simpl1l 989	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta ) -> ph )
Theorem	simpl1r 990	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta ) -> ps )
Theorem	simpl2l 991	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta ) -> ph )
Theorem	simpl2r 992	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta ) -> ps )
Theorem	simpl3l 993	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta ) -> ph )
Theorem	simpl3r 994	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta ) -> ps )
Theorem	simpr1l 995	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ph )
Theorem	simpr1r 996	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ps )
Theorem	simpr2l 997	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ph )
Theorem	simpr2r 998	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ps )
Theorem	simpr3l 999	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ph )
Theorem	simpr3r 1000	Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
|- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ps )