Type  Label  Description 
Statement 

Theorem  pm4.55dc 801 
Theorem *4.55 of [WhiteheadRussell] p.
120, for decidable propositions.
(Contributed by Jim Kingdon, 2May2018.)

DECID DECID 

Theorem  mpbiran 802 
Detach truth from conjunction in biconditional. (Contributed by NM,
27Feb1996.) (Revised by NM, 9Jan2015.)



Theorem  mpbiran2 803 
Detach truth from conjunction in biconditional. (Contributed by NM,
22Feb1996.) (Revised by NM, 9Jan2015.)



Theorem  mpbir2an 804 
Detach a conjunction of truths in a biconditional. (Contributed by NM,
10May2005.) (Revised by NM, 9Jan2015.)



Theorem  mpbi2and 805 
Detach a conjunction of truths in a biconditional. (Contributed by NM,
6Nov2011.) (Proof shortened by Wolf Lammen, 24Nov2012.)



Theorem  mpbir2and 806 
Detach a conjunction of truths in a biconditional. (Contributed by NM,
6Nov2011.) (Proof shortened by Wolf Lammen, 24Nov2012.)



Theorem  mpbiranOLD 807 
Obsolete version of mpbiran 802 as of 9Jan2015. (Contributed by NM,
27Feb1996.)



Theorem  mpbiran2OLD 808 
Obsolete version of mpbiran2 803 as of 9Jan2015. (Contributed by NM,
22Feb1996.)



Theorem  mpbir2anOLD 809 
Obsolete version of mpbir2an 804 as of 9Jan2015. (Contributed by NM,
10May2005.)



Theorem  mpbi2andOLD 810 
Obsolete version of mpbi2and 805 as of 9Jan2015. (Contributed by NM,
6Nov2011.) (Revised by NM, 24Nov2012.)



Theorem  mpbir2andOLD 811 
Obsolete version of mpbir2and 806 as of 9Jan2015. (Contributed by NM,
6Nov2011.) (Revised by NM, 24Nov2012.)



Theorem  pm5.62dc 812 
Theorem *5.62 of [WhiteheadRussell] p.
125, for a decidable proposition.
(Contributed by Jim Kingdon, 12May2018.)

DECID


Theorem  pm5.63dc 813 
Theorem *5.63 of [WhiteheadRussell] p.
125, for a decidable proposition.
(Contributed by Jim Kingdon, 12May2018.)

DECID 

Theorem  bianfi 814 
A wff conjoined with falsehood is false. (Contributed by NM,
5Aug1993.) (Proof shortened by Wolf Lammen, 26Nov2012.)



Theorem  bianfd 815 
A wff conjoined with falsehood is false. (Contributed by NM,
27Mar1995.) (Proof shortened by Wolf Lammen, 5Nov2013.)



Theorem  pm4.43 816 
Theorem *4.43 of [WhiteheadRussell] p.
119. (Contributed by NM,
3Jan2005.) (Proof shortened by Wolf Lammen, 26Nov2012.)



Theorem  pm4.82 817 
Theorem *4.82 of [WhiteheadRussell] p.
122. (Contributed by NM,
3Jan2005.)



Theorem  pm4.83dc 818 
Theorem *4.83 of [WhiteheadRussell] p.
122, for decidable propositions.
As with other case elimination theorems, like pm2.61dc 733, it only holds
for decidable propositions. (Contributed by Jim Kingdon, 12May2018.)

DECID


Theorem  biantr 819 
A transitive law of equivalence. Compare Theorem *4.22 of
[WhiteheadRussell] p. 117.
(Contributed by NM, 18Aug1993.)



Theorem  orbididc 820 
Disjunction distributes over the biconditional, for a decidable
proposition. Based on an axiom of system DS in Vladimir Lifschitz,
"On
calculational proofs" (1998),
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.3384.
(Contributed by Jim Kingdon, 2Apr2018.)

DECID 

Theorem  pm5.7dc 821 
Disjunction distributes over the biconditional, for a decidable
proposition. Based on theorem *5.7 of [WhiteheadRussell] p. 125. This
theorem is similar to orbididc 820. (Contributed by Jim Kingdon,
2Apr2018.)

DECID 

Theorem  bigolden 822 
DijkstraScholten's Golden Rule for calculational proofs. (Contributed by
NM, 10Jan2005.)



Theorem  anordc 823 
Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of
[WhiteheadRussell] p. 120, but
where the propositions are decidable. The
forward direction, pm3.1 648, holds for all propositions, but the
equivalence only holds given decidability. (Contributed by Jim Kingdon,
21Apr2018.)

DECID DECID


Theorem  pm3.11dc 824 
Theorem *3.11 of [WhiteheadRussell] p.
111, but for decidable
propositions. The converse, pm3.1 648, holds for all propositions, not
just decidable ones. (Contributed by Jim Kingdon, 22Apr2018.)

DECID DECID 

Theorem  pm3.12dc 825 
Theorem *3.12 of [WhiteheadRussell] p.
111, but for decidable
propositions. (Contributed by Jim Kingdon, 22Apr2018.)

DECID DECID 

Theorem  pm3.13dc 826 
Theorem *3.13 of [WhiteheadRussell] p.
111, but for decidable
propositions. The converse, pm3.14 647, holds for all propositions.
(Contributed by Jim Kingdon, 22Apr2018.)

DECID DECID 

Theorem  dn1dc 827 
DN_{1} for decidable propositions. Without the
decidability conditions,
DN_{1} can serve as a single axiom for
Boolean algebra. See
http://wwwunix.mcs.anl.gov/~mccune/papers/basax/v12.pdf.
(Contributed by Jim Kingdon, 22Apr2018.)

DECID DECID DECID DECID 

Theorem  pm5.71dc 828 
Decidable proposition version of theorem *5.71 of [WhiteheadRussell]
p. 125. (Contributed by Roy F. Longton, 23Jun2005.) (Modified for
decidability by Jim Kingdon, 19Apr2018.)

DECID 

Theorem  pm5.75 829 
Theorem *5.75 of [WhiteheadRussell] p.
126. (Contributed by NM,
3Jan2005.) (Proof shortened by Andrew Salmon, 7May2011.) (Proof
shortened by Wolf Lammen, 23Dec2012.)



Theorem  bimsc1 830 
Removal of conjunct from one side of an equivalence. (Contributed by NM,
5Aug1993.)



Theorem  ccase 831 
Inference for combining cases. (Contributed by NM, 29Jul1999.)
(Proof shortened by Wolf Lammen, 6Jan2013.)



Theorem  ccased 832 
Deduction for combining cases. (Contributed by NM, 9May2004.)



Theorem  ccase2 833 
Inference for combining cases. (Contributed by NM, 29Jul1999.)



Theorem  niabn 834 
Miscellaneous inference relating falsehoods. (Contributed by NM,
31Mar1994.)



Theorem  ninba 835 
Miscellaneous inference relating falsehoods. (Contributed by NM,
31Mar1994.)



Theorem  prlem1 836 
A specialized lemma for set theory (to derive the Axiom of Pairing).
(Contributed by NM, 18Oct1995.) (Proof shortened by Andrew Salmon,
13May2011.) (Proof shortened by Wolf Lammen, 5Jan2013.)



Theorem  prlem2 837 
A specialized lemma for set theory (to derive the Axiom of Pairing).
(Contributed by NM, 5Aug1993.) (Proof shortened by Andrew Salmon,
13May2011.) (Proof shortened by Wolf Lammen, 9Dec2012.)



Theorem  oplem1 838 
A specialized lemma for set theory (ordered pair theorem). (Contributed
by NM, 18Oct1995.) (Proof shortened by Wolf Lammen, 8Dec2012.)
(Proof shortened by Mario Carneiro, 2Feb2015.)



Theorem  rnlem 839 
Lemma used in construction of real numbers. (Contributed by NM,
4Sep1995.) (Proof shortened by Andrew Salmon, 26Jun2011.)



1.2.10 Abbreviated conjunction and disjunction of
three wff's


Syntax  w3o 840 
Extend wff definition to include 3way disjunction ('or').



Syntax  w3a 841 
Extend wff definition to include 3way conjunction ('and').



Definition  df3or 842 
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 661. (Contributed by NM,
8Apr1994.)



Definition  df3an 843 
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 379. (Contributed by NM,
8Apr1994.)



Theorem  3orass 844 
Associative law for triple disjunction. (Contributed by NM,
8Apr1994.)



Theorem  3anass 845 
Associative law for triple conjunction. (Contributed by NM,
8Apr1994.)



Theorem  3anrot 846 
Rotation law for triple conjunction. (Contributed by NM, 8Apr1994.)



Theorem  3orrot 847 
Rotation law for triple disjunction. (Contributed by NM, 4Apr1995.)



Theorem  3ancoma 848 
Commutation law for triple conjunction. (Contributed by NM,
21Apr1994.)



Theorem  3ancomb 849 
Commutation law for triple conjunction. (Contributed by NM,
21Apr1994.)



Theorem  3orcomb 850 
Commutation law for triple disjunction. (Contributed by Scott Fenton,
20Apr2011.)



Theorem  3anrev 851 
Reversal law for triple conjunction. (Contributed by NM, 21Apr1994.)



Theorem  3ioran 852 
Negated triple disjunction as triple conjunction. (Contributed by Scott
Fenton, 19Apr2011.)



Theorem  3simpa 853 
Simplification of triple conjunction. (Contributed by NM,
21Apr1994.)



Theorem  3simpb 854 
Simplification of triple conjunction. (Contributed by NM,
21Apr1994.)



Theorem  3simpc 855 
Simplification of triple conjunction. (Contributed by NM, 21Apr1994.)
(Proof shortened by Andrew Salmon, 13May2011.)



Theorem  simp1 856 
Simplification of triple conjunction. (Contributed by NM,
21Apr1994.)



Theorem  simp2 857 
Simplification of triple conjunction. (Contributed by NM,
21Apr1994.)



Theorem  simp3 858 
Simplification of triple conjunction. (Contributed by NM,
21Apr1994.)



Theorem  simpl1 859 
Simplification rule. (Contributed by Jeff Hankins, 17Nov2009.)



Theorem  simpl2 860 
Simplification rule. (Contributed by Jeff Hankins, 17Nov2009.)



Theorem  simpl3 861 
Simplification rule. (Contributed by Jeff Hankins, 17Nov2009.)



Theorem  simpr1 862 
Simplification rule. (Contributed by Jeff Hankins, 17Nov2009.)



Theorem  simpr2 863 
Simplification rule. (Contributed by Jeff Hankins, 17Nov2009.)



Theorem  simpr3 864 
Simplification rule. (Contributed by Jeff Hankins, 17Nov2009.)



Theorem  simp1i 865 
Infer a conjunct from a triple conjunction. (Contributed by NM,
19Apr2005.)



Theorem  simp2i 866 
Infer a conjunct from a triple conjunction. (Contributed by NM,
19Apr2005.)



Theorem  simp3i 867 
Infer a conjunct from a triple conjunction. (Contributed by NM,
19Apr2005.)



Theorem  simp1d 868 
Deduce a conjunct from a triple conjunction. (Contributed by NM,
4Sep2005.)



Theorem  simp2d 869 
Deduce a conjunct from a triple conjunction. (Contributed by NM,
4Sep2005.)



Theorem  simp3d 870 
Deduce a conjunct from a triple conjunction. (Contributed by NM,
4Sep2005.)



Theorem  simp1bi 871 
Deduce a conjunct from a triple conjunction. (Contributed by Jonathan
BenNaim, 3Jun2011.)



Theorem  simp2bi 872 
Deduce a conjunct from a triple conjunction. (Contributed by Jonathan
BenNaim, 3Jun2011.)



Theorem  simp3bi 873 
Deduce a conjunct from a triple conjunction. (Contributed by Jonathan
BenNaim, 3Jun2011.)



Theorem  3adant1 874 
Deduction adding a conjunct to antecedent. (Contributed by NM,
16Jul1995.)



Theorem  3adant2 875 
Deduction adding a conjunct to antecedent. (Contributed by NM,
16Jul1995.)



Theorem  3adant3 876 
Deduction adding a conjunct to antecedent. (Contributed by NM,
16Jul1995.)



Theorem  3ad2ant1 877 
Deduction adding conjuncts to an antecedent. (Contributed by NM,
21Apr2005.)



Theorem  3ad2ant2 878 
Deduction adding conjuncts to an antecedent. (Contributed by NM,
21Apr2005.)



Theorem  3ad2ant3 879 
Deduction adding conjuncts to an antecedent. (Contributed by NM,
21Apr2005.)



Theorem  simp1l 880 
Simplification of triple conjunction. (Contributed by NM, 9Nov2011.)



Theorem  simp1r 881 
Simplification of triple conjunction. (Contributed by NM, 9Nov2011.)



Theorem  simp2l 882 
Simplification of triple conjunction. (Contributed by NM, 9Nov2011.)



Theorem  simp2r 883 
Simplification of triple conjunction. (Contributed by NM, 9Nov2011.)



Theorem  simp3l 884 
Simplification of triple conjunction. (Contributed by NM, 9Nov2011.)



Theorem  simp3r 885 
Simplification of triple conjunction. (Contributed by NM, 9Nov2011.)



Theorem  simp11 886 
Simplification of doubly triple conjunction. (Contributed by NM,
17Nov2011.)



Theorem  simp12 887 
Simplification of doubly triple conjunction. (Contributed by NM,
17Nov2011.)



Theorem  simp13 888 
Simplification of doubly triple conjunction. (Contributed by NM,
17Nov2011.)



Theorem  simp21 889 
Simplification of doubly triple conjunction. (Contributed by NM,
17Nov2011.)



Theorem  simp22 890 
Simplification of doubly triple conjunction. (Contributed by NM,
17Nov2011.)



Theorem  simp23 891 
Simplification of doubly triple conjunction. (Contributed by NM,
17Nov2011.)



Theorem  simp31 892 
Simplification of doubly triple conjunction. (Contributed by NM,
17Nov2011.)



Theorem  simp32 893 
Simplification of doubly triple conjunction. (Contributed by NM,
17Nov2011.)



Theorem  simp33 894 
Simplification of doubly triple conjunction. (Contributed by NM,
17Nov2011.)



Theorem  simpll1 895 
Simplification of conjunction. (Contributed by NM, 9Mar2012.)



Theorem  simpll2 896 
Simplification of conjunction. (Contributed by NM, 9Mar2012.)



Theorem  simpll3 897 
Simplification of conjunction. (Contributed by NM, 9Mar2012.)



Theorem  simplr1 898 
Simplification of conjunction. (Contributed by NM, 9Mar2012.)



Theorem  simplr2 899 
Simplification of conjunction. (Contributed by NM, 9Mar2012.)



Theorem  simplr3 900 
Simplification of conjunction. (Contributed by NM, 9Mar2012.)

