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Theorem List for Intuitionistic Logic Explorer - 901-1000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempm3.11dc 901 Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 704, 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 902 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 903 Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 703, holds for all propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( -.  ( ph  /\  ps )  ->  ( -.  ph  \/  -.  ps ) ) ) )
 
Theoremdn1dc 904 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 905 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 ) ) ) )
 
Theorempm5.75 906 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 907 Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ( ph  ->  ps )  /\  ( ch 
 <->  ( ps  /\  ph )
 ) )  ->  ( ch 
 <-> 
 ph ) )
 
Theoremccase 908 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 909 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 910 Inference for combining cases. (Contributed by NM, 29-Jul-1999.)
 |-  ( ( ph  /\  ps )  ->  ta )   &    |-  ( ch  ->  ta )   &    |-  ( th  ->  ta )   =>    |-  ( ( ( ph  \/  ch )  /\  ( ps  \/  th ) ) 
 ->  ta )
 
Theoremniabn 911 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
 |-  ph   =>    |-  ( -.  ps  ->  ( ( ch  /\  ps ) 
 <->  -.  ph ) )
 
Theoremdedlem0a 912 Alternate version of dedlema 913. (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 913 Lemma for iftrue 3384. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  ( ph  ->  ( ps 
 <->  ( ( ps  /\  ph )  \/  ( ch 
 /\  -.  ph ) ) ) )
 
Theoremdedlemb 914 Lemma for iffalse 3387. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  ( -.  ph  ->  ( ch  <->  ( ( ps 
 /\  ph )  \/  ( ch  /\  -.  ph )
 ) ) )
 
Theorempm4.42r 915 One direction of Theorem *4.42 of [WhiteheadRussell] p. 119. (Contributed by Jim Kingdon, 4-Aug-2018.)
 |-  ( ( ( ph  /\ 
 ps )  \/  ( ph  /\  -.  ps )
 )  ->  ph )
 
Theoremninba 916 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
 |-  ph   =>    |-  ( -.  ps  ->  ( -.  ph  <->  ( ch  /\  ps ) ) )
 
Theoremprlem1 917 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 918 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 919 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 920 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
 
Syntaxw3o 921 Extend wff definition to include 3-way disjunction ('or').
 wff  ( ph  \/  ps  \/  ch )
 
Syntaxw3a 922 Extend wff definition to include 3-way conjunction ('and').
 wff  ( ph  /\  ps  /\ 
 ch )
 
Definitiondf-3or 923 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 717. (Contributed by NM, 8-Apr-1994.)
 |-  ( ( ph  \/  ps 
 \/  ch )  <->  ( ( ph  \/  ps )  \/  ch ) )
 
Definitiondf-3an 924 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 ) )
 
Theorem3orass 925 Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)
 |-  ( ( ph  \/  ps 
 \/  ch )  <->  ( ph  \/  ( ps  \/  ch )
 ) )
 
Theorem3anass 926 Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ph  /\  ( ps  /\  ch ) ) )
 
Theorem3anrot 927 Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ps  /\  ch 
 /\  ph ) )
 
Theorem3orrot 928 Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.)
 |-  ( ( ph  \/  ps 
 \/  ch )  <->  ( ps  \/  ch 
 \/  ph ) )
 
Theorem3ancoma 929 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ps  /\  ph 
 /\  ch ) )
 
Theorem3ancomb 930 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ph  /\  ch  /\ 
 ps ) )
 
Theorem3orcomb 931 Commutation law for triple disjunction. (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  ( ( ph  \/  ps 
 \/  ch )  <->  ( ph  \/  ch 
 \/  ps ) )
 
Theorem3anrev 932 Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ch  /\  ps 
 /\  ph ) )
 
Theorem3anan32 933 Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ( ph  /\ 
 ch )  /\  ps ) )
 
Theorem3anan12 934 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 935 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ( ph  /\ 
 ps )  /\  ( ph  /\  ch ) ) )
 
Theoremanandi3r 936 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ( ph  /\ 
 ps )  /\  ( ch  /\  ps ) ) )
 
Theorem3ioran 937 Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
 |-  ( -.  ( ph  \/  ps  \/  ch )  <->  ( -.  ph  /\  -.  ps  /\ 
 -.  ch ) )
 
Theorem3simpa 938 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ( ph  /\  ps ) )
 
Theorem3simpb 939 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ( ph  /\  ch ) )
 
Theorem3simpc 940 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ( ps  /\  ch ) )
 
Theoremsimp1 941 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ph )
 
Theoremsimp2 942 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ps )
 
Theoremsimp3 943 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ch )
 
Theoremsimpl1 944 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ph )
 
Theoremsimpl2 945 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ps )
 
Theoremsimpl3 946 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ch )
 
Theoremsimpr1 947 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  ps )
 
Theoremsimpr2 948 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  ch )
 
Theoremsimpr3 949 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  th )
 
Theoremsimp1i 950 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
 |-  ( ph  /\  ps  /\ 
 ch )   =>    |-  ph
 
Theoremsimp2i 951 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
 |-  ( ph  /\  ps  /\ 
 ch )   =>    |- 
 ps
 
Theoremsimp3i 952 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
 |-  ( ph  /\  ps  /\ 
 ch )   =>    |- 
 ch
 
Theoremsimp1d 953 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
 |-  ( ph  ->  ( ps  /\  ch  /\  th ) )   =>    |-  ( ph  ->  ps )
 
Theoremsimp2d 954 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
 |-  ( ph  ->  ( ps  /\  ch  /\  th ) )   =>    |-  ( ph  ->  ch )
 
Theoremsimp3d 955 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
 |-  ( ph  ->  ( ps  /\  ch  /\  th ) )   =>    |-  ( ph  ->  th )
 
Theoremsimp1bi 956 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  <->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ( ph  ->  ps )
 
Theoremsimp2bi 957 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  <->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ( ph  ->  ch )
 
Theoremsimp3bi 958 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  <->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ( ph  ->  th )
 
Theorem3adant1 959 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( th  /\  ph 
 /\  ps )  ->  ch )
 
Theorem3adant2 960 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  th  /\ 
 ps )  ->  ch )
 
Theorem3adant3 961 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ps  /\ 
 th )  ->  ch )
 
Theorem3ad2ant1 962 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
 |-  ( ph  ->  ch )   =>    |-  (
 ( ph  /\  ps  /\  th )  ->  ch )
 
Theorem3ad2ant2 963 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
 |-  ( ph  ->  ch )   =>    |-  (
 ( ps  /\  ph  /\  th )  ->  ch )
 
Theorem3ad2ant3 964 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
 |-  ( ph  ->  ch )   =>    |-  (
 ( ps  /\  th  /\  ph )  ->  ch )
 
Theoremsimp1l 965 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch  /\ 
 th )  ->  ph )
 
Theoremsimp1r 966 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch  /\ 
 th )  ->  ps )
 
Theoremsimp2l 967 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch )  /\  th )  ->  ps )
 
Theoremsimp2r 968 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch )  /\  th )  ->  ch )
 
Theoremsimp3l 969 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ph  /\  ps  /\  ( ch  /\  th ) )  ->  ch )
 
Theoremsimp3r 970 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ph  /\  ps  /\  ( ch  /\  th ) )  ->  th )
 
Theoremsimp11 971 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )  -> 
 ph )
 
Theoremsimp12 972 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )  ->  ps )
 
Theoremsimp13 973 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )  ->  ch )
 
Theoremsimp21 974 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th )  /\  ta )  ->  ps )
 
Theoremsimp22 975 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th )  /\  ta )  ->  ch )
 
Theoremsimp23 976 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th )  /\  ta )  ->  th )
 
Theoremsimp31 977 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ph  /\  ps  /\  ( ch  /\  th  /\ 
 ta ) )  ->  ch )
 
Theoremsimp32 978 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ph  /\  ps  /\  ( ch  /\  th  /\ 
 ta ) )  ->  th )
 
Theoremsimp33 979 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ph  /\  ps  /\  ( ch  /\  th  /\ 
 ta ) )  ->  ta )
 
Theoremsimpll1 980 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps  /\  ch )  /\  th )  /\  ta )  ->  ph )
 
Theoremsimpll2 981 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps  /\  ch )  /\  th )  /\  ta )  ->  ps )
 
Theoremsimpll3 982 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps  /\  ch )  /\  th )  /\  ta )  ->  ch )
 
Theoremsimplr1 983 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\  ( ph  /\  ps  /\ 
 ch ) )  /\  ta )  ->  ph )
 
Theoremsimplr2 984 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\  ( ph  /\  ps  /\ 
 ch ) )  /\  ta )  ->  ps )
 
Theoremsimplr3 985 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\  ( ph  /\  ps  /\ 
 ch ) )  /\  ta )  ->  ch )
 
Theoremsimprl1 986 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ( ph  /\  ps  /\ 
 ch )  /\  th ) )  ->  ph )
 
Theoremsimprl2 987 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ( ph  /\  ps  /\ 
 ch )  /\  th ) )  ->  ps )
 
Theoremsimprl3 988 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ( ph  /\  ps  /\ 
 ch )  /\  th ) )  ->  ch )
 
Theoremsimprr1 989 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )
 ) )  ->  ph )
 
Theoremsimprr2 990 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )
 ) )  ->  ps )
 
Theoremsimprr3 991 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )
 ) )  ->  ch )
 
Theoremsimpl1l 992 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps )  /\  ch  /\  th )  /\  ta )  ->  ph )
 
Theoremsimpl1r 993 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps )  /\  ch  /\  th )  /\  ta )  ->  ps )
 
Theoremsimpl2l 994 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( ch 
 /\  ( ph  /\  ps )  /\  th )  /\  ta )  ->  ph )
 
Theoremsimpl2r 995 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( ch 
 /\  ( ph  /\  ps )  /\  th )  /\  ta )  ->  ps )
 
Theoremsimpl3l 996 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( ch 
 /\  th  /\  ( ph  /\ 
 ps ) )  /\  ta )  ->  ph )
 
Theoremsimpl3r 997 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( ch 
 /\  th  /\  ( ph  /\ 
 ps ) )  /\  ta )  ->  ps )
 
Theoremsimpr1l 998 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ( ph  /\  ps )  /\  ch  /\  th ) )  ->  ph )
 
Theoremsimpr1r 999 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ( ph  /\  ps )  /\  ch  /\  th ) )  ->  ps )
 
Theoremsimpr2l 1000 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ch  /\  ( ph  /\ 
 ps )  /\  th ) )  ->  ph )
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