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Theorem List for Intuitionistic Logic Explorer - 901-1000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdcan 901 A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  -> DECID 
 ( ph  /\  ps )
 ) )
 
Theoremdcor 902 A disjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  -> DECID 
 ( ph  \/  ps )
 ) )
 
Theoremdcbi 903 An equivalence of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  -> DECID 
 ( ph  <->  ps ) ) )
 
Theoremannimdc 904 Express conjunction in terms of implication. The forward direction, annimim 658, is valid for all propositions, but as an equivalence, it requires a decidability condition. (Contributed by Jim Kingdon, 25-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( ( ph  /\ 
 -.  ps )  <->  -.  ( ph  ->  ps ) ) ) )
 
Theorempm4.55dc 905 Theorem *4.55 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 2-May-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( -.  ( -.  ph  /\  ps )  <->  (
 ph  \/  -.  ps )
 ) ) )
 
Theoremorandc 906 Disjunction in terms of conjunction (De Morgan's law), for decidable propositions. Compare Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by Jim Kingdon, 13-Dec-2021.)
 |-  ( (DECID 
 ph  /\ DECID  ps )  ->  (
 ( ph  \/  ps )  <->  -.  ( -.  ph  /\  -.  ps ) ) )
 
Theoremmpbiran 907 Detach truth from conjunction in biconditional. (Contributed by NM, 27-Feb-1996.) (Revised by NM, 9-Jan-2015.)
 |- 
 ps   &    |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ph  <->  ch )
 
Theoremmpbiran2 908 Detach truth from conjunction in biconditional. (Contributed by NM, 22-Feb-1996.) (Revised by NM, 9-Jan-2015.)
 |- 
 ch   &    |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ph  <->  ps )
 
Theoremmpbir2an 909 Detach a conjunction of truths in a biconditional. (Contributed by NM, 10-May-2005.) (Revised by NM, 9-Jan-2015.)
 |- 
 ps   &    |- 
 ch   &    |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ph
 
Theoremmpbi2and 910 Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( ( ps  /\  ch ) 
 <-> 
 th ) )   =>    |-  ( ph  ->  th )
 
Theoremmpbir2and 911 Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
 |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ( ps  <->  ( ch  /\  th ) ) )   =>    |-  ( ph  ->  ps )
 
Theorempm5.62dc 912 Theorem *5.62 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
 |-  (DECID 
 ps  ->  ( ( (
 ph  /\  ps )  \/  -.  ps )  <->  ( ph  \/  -. 
 ps ) ) )
 
Theorempm5.63dc 913 Theorem *5.63 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
 |-  (DECID 
 ph  ->  ( ( ph  \/  ps )  <->  ( ph  \/  ( -.  ph  /\  ps )
 ) ) )
 
Theorembianfi 914 A wff conjoined with falsehood is false. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)
 |- 
 -.  ph   =>    |-  ( ph  <->  ( ps  /\  ph ) )
 
Theorembianfd 915 A wff conjoined with falsehood is false. (Contributed by NM, 27-Mar-1995.) (Proof shortened by Wolf Lammen, 5-Nov-2013.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  ( ps 
 <->  ( ps  /\  ch ) ) )
 
Theorempm4.43 916 Theorem *4.43 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)
 |-  ( ph  <->  ( ( ph  \/  ps )  /\  ( ph  \/  -.  ps )
 ) )
 
Theorempm4.82 917 Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  -.  ps )
 ) 
 <->  -.  ph )
 
Theorempm4.83dc 918 Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions. As with other case elimination theorems, like pm2.61dc 833, it only holds for decidable propositions. (Contributed by Jim Kingdon, 12-May-2018.)
 |-  (DECID 
 ph  ->  ( ( (
 ph  ->  ps )  /\  ( -.  ph  ->  ps )
 ) 
 <->  ps ) )
 
Theorembiantr 919 A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.)
 |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  ps ) )  ->  ( ph  <->  ch ) )
 
Theoremorbididc 920 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, 2-Apr-2018.)
 |-  (DECID 
 ph  ->  ( ( ph  \/  ( ps  <->  ch ) )  <->  ( ( ph  \/  ps )  <->  ( ph  \/  ch ) ) ) )
 
Theorempm5.7dc 921 Disjunction distributes over the biconditional, for a decidable proposition. Based on theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbididc 920. (Contributed by Jim Kingdon, 2-Apr-2018.)
 |-  (DECID 
 ch  ->  ( ( (
 ph  \/  ch )  <->  ( ps  \/  ch )
 ) 
 <->  ( ch  \/  ( ph 
 <->  ps ) ) ) )
 
Theorembigolden 922 Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.)
 |-  ( ( ( ph  /\ 
 ps )  <->  ph )  <->  ( ps  <->  ( ph  \/  ps ) ) )
 
Theoremanordc 923 Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120, but where the propositions are decidable. The forward direction, pm3.1 726, holds for all propositions, but the equivalence only holds given decidability. (Contributed by Jim Kingdon, 21-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( ( ph  /\ 
 ps )  <->  -.  ( -.  ph  \/  -.  ps ) ) ) )
 
Theorempm3.11dc 924 Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 726, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 22-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( -.  ( -.  ph  \/  -.  ps )  ->  ( ph  /\  ps ) ) ) )
 
Theorempm3.12dc 925 Theorem *3.12 of [WhiteheadRussell] p. 111, but for decidable propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( ( -.  ph  \/  -.  ps )  \/  ( ph  /\  ps ) ) ) )
 
Theorempm3.13dc 926 Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 725, holds for all propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( -.  ( ph  /\  ps )  ->  ( -.  ph  \/  -.  ps ) ) ) )
 
Theoremdn1dc 927 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 ) )
 
Theorempm5.71dc 928 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 ) ) ) )
 
Theorempm5.75 929 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 ) )
 
Theorembimsc1 930 Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ( ph  ->  ps )  /\  ( ch 
 <->  ( ps  /\  ph )
 ) )  ->  ( ch 
 <-> 
 ph ) )
 
Theoremccase 931 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 )
 
Theoremccased 932 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 ) )
 
Theoremccase2 933 Inference for combining cases. (Contributed by NM, 29-Jul-1999.)
 |-  ( ( ph  /\  ps )  ->  ta )   &    |-  ( ch  ->  ta )   &    |-  ( th  ->  ta )   =>    |-  ( ( ( ph  \/  ch )  /\  ( ps  \/  th ) ) 
 ->  ta )
 
Theoremniabn 934 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
 |-  ph   =>    |-  ( -.  ps  ->  ( ( ch  /\  ps ) 
 <->  -.  ph ) )
 
Theoremdedlem0a 935 Alternate version of dedlema 936. (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 ) ) ) )
 
Theoremdedlema 936 Lemma for iftrue 3447. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  ( ph  ->  ( ps 
 <->  ( ( ps  /\  ph )  \/  ( ch 
 /\  -.  ph ) ) ) )
 
Theoremdedlemb 937 Lemma for iffalse 3450. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  ( -.  ph  ->  ( ch  <->  ( ( ps 
 /\  ph )  \/  ( ch  /\  -.  ph )
 ) ) )
 
Theorempm4.42r 938 One direction of Theorem *4.42 of [WhiteheadRussell] p. 119. (Contributed by Jim Kingdon, 4-Aug-2018.)
 |-  ( ( ( ph  /\ 
 ps )  \/  ( ph  /\  -.  ps )
 )  ->  ph )
 
Theoremninba 939 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
 |-  ph   =>    |-  ( -.  ps  ->  ( -.  ph  <->  ( ch  /\  ps ) ) )
 
Theoremprlem1 940 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 ) ) )
 
Theoremprlem2 941 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 ) ) ) )
 
Theoremoplem1 942 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 )
 
Theoremrnlem 943 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 ) ) ) )
 
1.2.11  Abbreviated conjunction and disjunction of three wff's
 
Syntaxw3o 944 Extend wff definition to include 3-way disjunction ('or').
 wff  ( ph  \/  ps  \/  ch )
 
Syntaxw3a 945 Extend wff definition to include 3-way conjunction ('and').
 wff  ( ph  /\  ps  /\ 
 ch )
 
Definitiondf-3or 946 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 739. (Contributed by NM, 8-Apr-1994.)
 |-  ( ( ph  \/  ps 
 \/  ch )  <->  ( ( ph  \/  ps )  \/  ch ) )
 
Definitiondf-3an 947 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 396. (Contributed by NM, 8-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ( ph  /\ 
 ps )  /\  ch ) )
 
Theorem3orass 948 Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)
 |-  ( ( ph  \/  ps 
 \/  ch )  <->  ( ph  \/  ( ps  \/  ch )
 ) )
 
Theorem3anass 949 Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ph  /\  ( ps  /\  ch ) ) )
 
Theorem3anrot 950 Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ps  /\  ch 
 /\  ph ) )
 
Theorem3orrot 951 Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.)
 |-  ( ( ph  \/  ps 
 \/  ch )  <->  ( ps  \/  ch 
 \/  ph ) )
 
Theorem3ancoma 952 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ps  /\  ph 
 /\  ch ) )
 
Theorem3ancomb 953 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ph  /\  ch  /\ 
 ps ) )
 
Theorem3orcomb 954 Commutation law for triple disjunction. (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  ( ( ph  \/  ps 
 \/  ch )  <->  ( ph  \/  ch 
 \/  ps ) )
 
Theorem3anrev 955 Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ch  /\  ps 
 /\  ph ) )
 
Theorem3anan32 956 Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ( ph  /\ 
 ch )  /\  ps ) )
 
Theorem3anan12 957 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 )
 ) )
 
Theoremanandi3 958 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ( ph  /\ 
 ps )  /\  ( ph  /\  ch ) ) )
 
Theoremanandi3r 959 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ( ph  /\ 
 ps )  /\  ( ch  /\  ps ) ) )
 
Theorem3ioran 960 Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
 |-  ( -.  ( ph  \/  ps  \/  ch )  <->  ( -.  ph  /\  -.  ps  /\ 
 -.  ch ) )
 
Theorem3simpa 961 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ( ph  /\  ps ) )
 
Theorem3simpb 962 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ( ph  /\  ch ) )
 
Theorem3simpc 963 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ( ps  /\  ch ) )
 
Theoremsimp1 964 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ph )
 
Theoremsimp2 965 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ps )
 
Theoremsimp3 966 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ch )
 
Theoremsimpl1 967 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ph )
 
Theoremsimpl2 968 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ps )
 
Theoremsimpl3 969 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ch )
 
Theoremsimpr1 970 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  ps )
 
Theoremsimpr2 971 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  ch )
 
Theoremsimpr3 972 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  th )
 
Theoremsimp1i 973 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
 |-  ( ph  /\  ps  /\ 
 ch )   =>    |-  ph
 
Theoremsimp2i 974 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
 |-  ( ph  /\  ps  /\ 
 ch )   =>    |- 
 ps
 
Theoremsimp3i 975 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
 |-  ( ph  /\  ps  /\ 
 ch )   =>    |- 
 ch
 
Theoremsimp1d 976 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
 |-  ( ph  ->  ( ps  /\  ch  /\  th ) )   =>    |-  ( ph  ->  ps )
 
Theoremsimp2d 977 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
 |-  ( ph  ->  ( ps  /\  ch  /\  th ) )   =>    |-  ( ph  ->  ch )
 
Theoremsimp3d 978 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
 |-  ( ph  ->  ( ps  /\  ch  /\  th ) )   =>    |-  ( ph  ->  th )
 
Theoremsimp1bi 979 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  <->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ( ph  ->  ps )
 
Theoremsimp2bi 980 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  <->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ( ph  ->  ch )
 
Theoremsimp3bi 981 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  <->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ( ph  ->  th )
 
Theorem3adant1 982 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( th  /\  ph 
 /\  ps )  ->  ch )
 
Theorem3adant2 983 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  th  /\ 
 ps )  ->  ch )
 
Theorem3adant3 984 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ps  /\ 
 th )  ->  ch )
 
Theorem3ad2ant1 985 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
 |-  ( ph  ->  ch )   =>    |-  (
 ( ph  /\  ps  /\  th )  ->  ch )
 
Theorem3ad2ant2 986 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
 |-  ( ph  ->  ch )   =>    |-  (
 ( ps  /\  ph  /\  th )  ->  ch )
 
Theorem3ad2ant3 987 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
 |-  ( ph  ->  ch )   =>    |-  (
 ( ps  /\  th  /\  ph )  ->  ch )
 
Theoremsimp1l 988 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch  /\ 
 th )  ->  ph )
 
Theoremsimp1r 989 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch  /\ 
 th )  ->  ps )
 
Theoremsimp2l 990 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch )  /\  th )  ->  ps )
 
Theoremsimp2r 991 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch )  /\  th )  ->  ch )
 
Theoremsimp3l 992 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ph  /\  ps  /\  ( ch  /\  th ) )  ->  ch )
 
Theoremsimp3r 993 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ph  /\  ps  /\  ( ch  /\  th ) )  ->  th )
 
Theoremsimp11 994 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )  -> 
 ph )
 
Theoremsimp12 995 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )  ->  ps )
 
Theoremsimp13 996 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )  ->  ch )
 
Theoremsimp21 997 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th )  /\  ta )  ->  ps )
 
Theoremsimp22 998 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th )  /\  ta )  ->  ch )
 
Theoremsimp23 999 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th )  /\  ta )  ->  th )
 
Theoremsimp31 1000 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ph  /\  ps  /\  ( ch  /\  th  /\ 
 ta ) )  ->  ch )
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