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Theorem List for Metamath Proof Explorer - 901-1000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempclem6 901 Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Nov-2012.)
 |-  ( ( ph  <->  ( ps  /\  -.  ph ) )  ->  -.  ps )
 
Theorembiantr 902 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 ) )
 
Theoremorbidi 903 Disjunction distributes over the biconditional. 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 NM, 8-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Feb-2013.)
 |-  ( ( ph  \/  ( ps  <->  ch ) )  <->  ( ( ph  \/  ps )  <->  ( ph  \/  ch ) ) )
 
Theorembiluk 904 Lukasiewicz's shortest axiom for equivalential calculus. Storrs McCall, ed., Polish Logic 1920-1939 (Oxford, 1967), p. 96. (Contributed by NM, 10-Jan-2005.)
 |-  ( ( ph  <->  ps )  <->  ( ( ch  <->  ps )  <->  ( ph  <->  ch ) ) )
 
Theorempm5.7 905 Disjunction distributes over the biconditional. Theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbidi 903. (Contributed by Roy F. Longton, 21-Jun-2005.)
 |-  ( ( ( ph  \/  ch )  <->  ( ps  \/  ch ) )  <->  ( ch  \/  ( ph  <->  ps ) ) )
 
Theorembigolden 906 Dijkstra-Scholten's Golden Rule for calculational proofs. (Contributed by NM, 10-Jan-2005.)
 |-  ( ( ( ph  /\ 
 ps )  <->  ph )  <->  ( ps  <->  ( ph  \/  ps ) ) )
 
Theorempm5.71 907 Theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.)
 |-  ( ( ps  ->  -. 
 ch )  ->  (
 ( ( ph  \/  ps )  /\  ch )  <->  (
 ph  /\  ch )
 ) )
 
Theorempm5.75 908 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 909 Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ( ph  ->  ps )  /\  ( ch 
 <->  ( ps  /\  ph )
 ) )  ->  ( ch 
 <-> 
 ph ) )
 
Theorem4exmid 910 The disjunction of the four possible combinations of two wffs and their negations is always true. (Contributed by David Abernethy, 28-Jan-2014.)
 |-  ( ( ( ph  /\ 
 ps )  \/  ( -.  ph  /\  -.  ps ) )  \/  (
 ( ph  /\  -.  ps )  \/  ( ps  /\  -.  ph ) ) )
 
Theoremecase2d 911 Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Dec-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  -.  ( ps  /\ 
 ch ) )   &    |-  ( ph  ->  -.  ( ps  /\ 
 th ) )   &    |-  ( ph  ->  ( ta  \/  ( ch  \/  th )
 ) )   =>    |-  ( ph  ->  ta )
 
Theoremecase3 912 Inference for elimination by cases. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)
 |-  ( ph  ->  ch )   &    |-  ( ps  ->  ch )   &    |-  ( -.  ( ph  \/  ps )  ->  ch )   =>    |- 
 ch
 
Theoremecase 913 Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.)
 |-  ( -.  ph  ->  ch )   &    |-  ( -.  ps  ->  ch )   &    |-  ( ( ph  /\ 
 ps )  ->  ch )   =>    |-  ch
 
Theoremecase3d 914 Deduction for elimination by cases. (Contributed by NM, 2-May-1996.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  ( ph  ->  ( ps  ->  th ) )   &    |-  ( ph  ->  ( ch  ->  th ) )   &    |-  ( ph  ->  ( -.  ( ps  \/  ch )  ->  th )
 )   =>    |-  ( ph  ->  th )
 
Theoremecased 915 Deduction for elimination by cases. (Contributed by NM, 8-Oct-2012.)
 |-  ( ph  ->  ( -.  ps  ->  th )
 )   &    |-  ( ph  ->  ( -.  ch  ->  th )
 )   &    |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ph  ->  th )
 
Theoremecase3ad 916 Deduction for elimination by cases. (Contributed by NM, 24-May-2013.)
 |-  ( ph  ->  ( ps  ->  th ) )   &    |-  ( ph  ->  ( ch  ->  th ) )   &    |-  ( ph  ->  ( ( -.  ps  /\  -. 
 ch )  ->  th )
 )   =>    |-  ( ph  ->  th )
 
Theoremccase 917 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 918 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 919 Inference for combining cases. (Contributed by NM, 29-Jul-1999.)
 |-  ( ( ph  /\  ps )  ->  ta )   &    |-  ( ch  ->  ta )   &    |-  ( th  ->  ta )   =>    |-  ( ( ( ph  \/  ch )  /\  ( ps  \/  th ) ) 
 ->  ta )
 
Theorem4cases 920 Inference eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 25-Oct-2003.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 -.  ps )  ->  ch )   &    |-  (
 ( -.  ph  /\  ps )  ->  ch )   &    |-  ( ( -.  ph  /\  -.  ps )  ->  ch )   =>    |- 
 ch
 
Theorem4casesdan 921 Deduction eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 19-Mar-2013.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   &    |-  ( ( ph  /\  ( ps  /\  -.  ch ) )  ->  th )   &    |-  (
 ( ph  /\  ( -. 
 ps  /\  ch )
 )  ->  th )   &    |-  (
 ( ph  /\  ( -. 
 ps  /\  -.  ch )
 )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremniabn 922 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
 |-  ph   =>    |-  ( -.  ps  ->  ( ( ch  /\  ps ) 
 <->  -.  ph ) )
 
Theoremdedlem0a 923 Lemma for an alternate version of weak deduction theorem. (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 ) ) ) )
 
Theoremdedlem0b 924 Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.)
 |-  ( -.  ph  ->  ( ps  <->  ( ( ps 
 ->  ph )  ->  ( ch  /\  ph ) ) ) )
 
Theoremdedlema 925 Lemma for weak deduction theorem. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  ( ph  ->  ( ps 
 <->  ( ( ps  /\  ph )  \/  ( ch 
 /\  -.  ph ) ) ) )
 
Theoremdedlemb 926 Lemma for weak deduction theorem. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  ( -.  ph  ->  ( ch  <->  ( ( ps 
 /\  ph )  \/  ( ch  /\  -.  ph )
 ) ) )
 
Theoremelimh 927 Hypothesis builder for weak deduction theorem. For more information, see the Deduction Theorem link on the Metamath Proof Explorer home page. (Contributed by NM, 26-Jun-2002.)
 |-  ( ( ph  <->  ( ( ph  /\ 
 ch )  \/  ( ps  /\  -.  ch )
 ) )  ->  ( ch 
 <->  ta ) )   &    |-  (
 ( ps  <->  ( ( ph  /\ 
 ch )  \/  ( ps  /\  -.  ch )
 ) )  ->  ( th 
 <->  ta ) )   &    |-  th   =>    |-  ta
 
Theoremdedt 928 The weak deduction theorem. For more information, see the Deduction Theorem link on the Metamath Proof Explorer home page. (Contributed by NM, 26-Jun-2002.)
 |-  ( ( ph  <->  ( ( ph  /\ 
 ch )  \/  ( ps  /\  -.  ch )
 ) )  ->  ( th 
 <->  ta ) )   &    |-  ta   =>    |-  ( ch  ->  th )
 
Theoremcon3th 929 Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. This version of con3 128 demonstrates the use of the weak deduction theorem dedt 928 to derive it from con3i 129. (Contributed by NM, 27-Jun-2002.) (Proof modification is discouraged.)
 |-  ( ( ph  ->  ps )  ->  ( -.  ps 
 ->  -.  ph ) )
 
Theoremconsensus 930 The consensus theorem. This theorem and its dual (with  \/ and  /\ interchanged) are commonly used in computer logic design to eliminate redundant terms from Boolean expressions. Specifically, we prove that the term  ( ps  /\  ch ) on the left-hand side is redundant. (Contributed by NM, 16-May-2003.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 20-Jan-2013.)
 |-  ( ( ( (
 ph  /\  ps )  \/  ( -.  ph  /\  ch ) )  \/  ( ps  /\  ch ) )  <-> 
 ( ( ph  /\  ps )  \/  ( -.  ph  /\ 
 ch ) ) )
 
Theorempm4.42 931 Theorem *4.42 of [WhiteheadRussell] p. 119. (Contributed by Roy F. Longton, 21-Jun-2005.)
 |-  ( ph  <->  ( ( ph  /\ 
 ps )  \/  ( ph  /\  -.  ps )
 ) )
 
Theoremninba 932 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
 |-  ph   =>    |-  ( -.  ps  ->  ( -.  ph  <->  ( ch  /\  ps ) ) )
 
Theoremprlem1 933 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 934 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 935 A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.)
 |-  ( ph  ->  ( ps  \/  ch ) )   &    |-  ( ph  ->  ( th  \/  ta ) )   &    |-  ( ps 
 <-> 
 th )   &    |-  ( ch  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ps )
 
Theoremrnlem 936 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 ) ) ) )
 
Theoremdn1 937 A single axiom for Boolean algebra known as DN1. See http://www-unix.mcs.anl.gov/~mccune/papers/basax/v12.pdf. (Contributed by Jeffrey Hankins, 3-Jul-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)
 |-  ( -.  ( -.  ( -.  ( ph  \/  ps )  \/  ch )  \/  -.  ( ph  \/  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) )  <->  ch )
 
1.3.8  Abbreviated conjunction and disjunction of three wff's
 
Syntaxw3o 938 Extend wff definition to include 3-way disjunction ('or').
 wff  ( ph  \/  ps  \/  ch )
 
Syntaxw3a 939 Extend wff definition to include 3-way conjunction ('and').
 wff  ( ph  /\  ps  /\ 
 ch )
 
Definitiondf-3or 940 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 512. (Contributed by NM, 8-Apr-1994.)
 |-  ( ( ph  \/  ps 
 \/  ch )  <->  ( ( ph  \/  ps )  \/  ch ) )
 
Definitiondf-3an 941 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 633. (Contributed by NM, 8-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ( ph  /\ 
 ps )  /\  ch ) )
 
Theorem3orass 942 Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)
 |-  ( ( ph  \/  ps 
 \/  ch )  <->  ( ph  \/  ( ps  \/  ch )
 ) )
 
Theorem3anass 943 Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ph  /\  ( ps  /\  ch ) ) )
 
Theorem3anrot 944 Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ps  /\  ch 
 /\  ph ) )
 
Theorem3orrot 945 Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.)
 |-  ( ( ph  \/  ps 
 \/  ch )  <->  ( ps  \/  ch 
 \/  ph ) )
 
Theorem3ancoma 946 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ps  /\  ph 
 /\  ch ) )
 
Theorem3orcoma 947 Commutation law for triple disjunction. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ( ph  \/  ps 
 \/  ch )  <->  ( ps  \/  ph 
 \/  ch ) )
 
Theorem3ancomb 948 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ph  /\  ch  /\ 
 ps ) )
 
Theorem3orcomb 949 Commutation law for triple disjunction. (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  ( ( ph  \/  ps 
 \/  ch )  <->  ( ph  \/  ch 
 \/  ps ) )
 
Theorem3anrev 950 Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ch  /\  ps 
 /\  ph ) )
 
Theorem3anan32 951 Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  ( ( ph  /\ 
 ch )  /\  ps ) )
 
Theorem3anan12 952 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 )
 ) )
 
Theorem3anor 953 Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  <->  -.  ( -.  ph  \/  -.  ps  \/  -.  ch ) )
 
Theorem3ianor 954 Negated triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( -.  ( ph  /\ 
 ps  /\  ch )  <->  ( -.  ph  \/  -.  ps  \/  -.  ch ) )
 
Theorem3ioran 955 Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
 |-  ( -.  ( ph  \/  ps  \/  ch )  <->  ( -.  ph  /\  -.  ps  /\ 
 -.  ch ) )
 
Theorem3oran 956 Triple disjunction in terms of triple conjunction. (Contributed by NM, 8-Oct-2012.)
 |-  ( ( ph  \/  ps 
 \/  ch )  <->  -.  ( -.  ph  /\ 
 -.  ps  /\  -.  ch ) )
 
Theorem3simpa 957 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ( ph  /\  ps ) )
 
Theorem3simpb 958 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ( ph  /\  ch ) )
 
Theorem3simpc 959 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ( ps  /\  ch ) )
 
Theoremsimp1 960 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ph )
 
Theoremsimp2 961 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ps )
 
Theoremsimp3 962 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ch )
 
Theoremsimpl1 963 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ph )
 
Theoremsimpl2 964 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ps )
 
Theoremsimpl3 965 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ch )
 
Theoremsimpr1 966 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  ps )
 
Theoremsimpr2 967 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  ch )
 
Theoremsimpr3 968 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  th )
 
Theoremsimp1i 969 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
 |-  ( ph  /\  ps  /\ 
 ch )   =>    |-  ph
 
Theoremsimp2i 970 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
 |-  ( ph  /\  ps  /\ 
 ch )   =>    |- 
 ps
 
Theoremsimp3i 971 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
 |-  ( ph  /\  ps  /\ 
 ch )   =>    |- 
 ch
 
Theoremsimp1d 972 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
 |-  ( ph  ->  ( ps  /\  ch  /\  th ) )   =>    |-  ( ph  ->  ps )
 
Theoremsimp2d 973 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
 |-  ( ph  ->  ( ps  /\  ch  /\  th ) )   =>    |-  ( ph  ->  ch )
 
Theoremsimp3d 974 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
 |-  ( ph  ->  ( ps  /\  ch  /\  th ) )   =>    |-  ( ph  ->  th )
 
Theoremsimp1bi 975 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  <->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ( ph  ->  ps )
 
Theoremsimp2bi 976 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  <->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ( ph  ->  ch )
 
Theoremsimp3bi 977 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  <->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ( ph  ->  th )
 
Theorem3adant1 978 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( th  /\  ph 
 /\  ps )  ->  ch )
 
Theorem3adant2 979 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  th  /\ 
 ps )  ->  ch )
 
Theorem3adant3 980 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ps  /\ 
 th )  ->  ch )
 
Theorem3ad2ant1 981 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
 |-  ( ph  ->  ch )   =>    |-  (
 ( ph  /\  ps  /\  th )  ->  ch )
 
Theorem3ad2ant2 982 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
 |-  ( ph  ->  ch )   =>    |-  (
 ( ps  /\  ph  /\  th )  ->  ch )
 
Theorem3ad2ant3 983 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
 |-  ( ph  ->  ch )   =>    |-  (
 ( ps  /\  th  /\  ph )  ->  ch )
 
Theoremsimp1l 984 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch  /\ 
 th )  ->  ph )
 
Theoremsimp1r 985 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch  /\ 
 th )  ->  ps )
 
Theoremsimp2l 986 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch )  /\  th )  ->  ps )
 
Theoremsimp2r 987 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch )  /\  th )  ->  ch )
 
Theoremsimp3l 988 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ph  /\  ps  /\  ( ch  /\  th ) )  ->  ch )
 
Theoremsimp3r 989 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
 |-  ( ( ph  /\  ps  /\  ( ch  /\  th ) )  ->  th )
 
Theoremsimp11 990 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )  -> 
 ph )
 
Theoremsimp12 991 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )  ->  ps )
 
Theoremsimp13 992 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )  ->  ch )
 
Theoremsimp21 993 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th )  /\  ta )  ->  ps )
 
Theoremsimp22 994 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th )  /\  ta )  ->  ch )
 
Theoremsimp23 995 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th )  /\  ta )  ->  th )
 
Theoremsimp31 996 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ph  /\  ps  /\  ( ch  /\  th  /\ 
 ta ) )  ->  ch )
 
Theoremsimp32 997 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ph  /\  ps  /\  ( ch  /\  th  /\ 
 ta ) )  ->  th )
 
Theoremsimp33 998 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
 |-  ( ( ph  /\  ps  /\  ( ch  /\  th  /\ 
 ta ) )  ->  ta )
 
Theoremsimpll1 999 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps  /\  ch )  /\  th )  /\  ta )  ->  ph )
 
Theoremsimpll2 1000 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps  /\  ch )  /\  th )  /\  ta )  ->  ps )
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