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Theorem List for Metamath Proof Explorer - 801-900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoreman42 801 Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) )  <-> 
 ( ( ph  /\  ch )  /\  ( th  /\  ps ) ) )
 
Theoreman4s 802 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  ( ps  /\  th ) ) 
 ->  ta )
 
Theoreman42s 803 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  ( th  /\  ps ) ) 
 ->  ta )
 
Theoremanandi 804 Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
 |-  ( ( ph  /\  ( ps  /\  ch ) )  <-> 
 ( ( ph  /\  ps )  /\  ( ph  /\  ch ) ) )
 
Theoremanandir 805 Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch ) 
 <->  ( ( ph  /\  ch )  /\  ( ps  /\  ch ) ) )
 
Theoremanandis 806 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ph  /\  ch ) ) 
 ->  ta )   =>    |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  ta )
 
Theoremanandirs 807 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
 |-  ( ( ( ph  /\ 
 ch )  /\  ( ps  /\  ch ) ) 
 ->  ta )   =>    |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  ta )
 
Theoremimpbida 808 Deduce an equivalence from two implications. (Contributed by NM, 17-Feb-2007.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 ch )  ->  ps )   =>    |-  ( ph  ->  ( ps  <->  ch ) )
 
Theorempm3.48 809 Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM, 28-Jan-1997.)
 |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  th ) )  ->  ( ( ph  \/  ch )  ->  ( ps  \/  th ) ) )
 
Theorempm3.45 810 Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ps )  ->  ( ( ph  /\  ch )  ->  ( ps  /\  ch )
 ) )
 
Theoremim2anan9 811 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ta  ->  et ) )   =>    |-  ( ( ph  /\  th )  ->  ( ( ps 
 /\  ta )  ->  ( ch  /\  et ) ) )
 
Theoremim2anan9r 812 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ta  ->  et ) )   =>    |-  ( ( th  /\  ph )  ->  ( ( ps  /\  ta )  ->  ( ch  /\  et )
 ) )
 
Theoremanim12dan 813 Conjoin antecedents and consequents in a deduction. (Contributed by Mario Carneiro, 12-May-2014.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 th )  ->  ta )   =>    |-  (
 ( ph  /\  ( ps 
 /\  th ) )  ->  ( ch  /\  ta )
 )
 
Theoremorim12d 814 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 10-May-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ta ) )   =>    |-  ( ph  ->  (
 ( ps  \/  th )  ->  ( ch  \/  ta ) ) )
 
Theoremorim1d 815 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( ps  \/  th )  ->  ( ch  \/  th ) ) )
 
Theoremorim2d 816 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( th  \/  ps )  ->  ( th  \/  ch ) ) )
 
Theoremorim2 817 Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ps  ->  ch )  ->  ( ( ph  \/  ps )  ->  ( ph  \/  ch )
 ) )
 
Theorempm2.38 818 Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
 |-  ( ( ps  ->  ch )  ->  ( ( ps  \/  ph )  ->  ( ch  \/  ph ) ) )
 
Theorempm2.36 819 Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
 |-  ( ( ps  ->  ch )  ->  ( ( ph  \/  ps )  ->  ( ch  \/  ph )
 ) )
 
Theorempm2.37 820 Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
 |-  ( ( ps  ->  ch )  ->  ( ( ps  \/  ph )  ->  ( ph  \/  ch ) ) )
 
Theorempm2.73 821 Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ps )  ->  ( (
 ( ph  \/  ps )  \/  ch )  ->  ( ps  \/  ch ) ) )
 
Theorempm2.74 822 Theorem *2.74 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  ( ( ps  ->  ph )  ->  ( (
 ( ph  \/  ps )  \/  ch )  ->  ( ph  \/  ch ) ) )
 
Theoremorimdi 823 Disjunction distributes over implication. (Contributed by Wolf Lammen, 5-Jan-2013.)
 |-  ( ( ph  \/  ( ps  ->  ch )
 ) 
 <->  ( ( ph  \/  ps )  ->  ( ph  \/  ch ) ) )
 
Theorempm2.76 824 Theorem *2.76 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ( ps  ->  ch )
 )  ->  ( ( ph  \/  ps )  ->  ( ph  \/  ch )
 ) )
 
Theorempm2.75 825 Theorem *2.75 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Jan-2013.)
 |-  ( ( ph  \/  ps )  ->  ( ( ph  \/  ( ps  ->  ch ) )  ->  ( ph  \/  ch ) ) )
 
Theorempm2.8 826 Theorem *2.8 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)
 |-  ( ( ph  \/  ps )  ->  ( ( -.  ps  \/  ch )  ->  ( ph  \/  ch ) ) )
 
Theorempm2.81 827 Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ps  ->  ( ch  ->  th )
 )  ->  ( ( ph  \/  ps )  ->  ( ( ph  \/  ch )  ->  ( ph  \/  th ) ) ) )
 
Theorempm2.82 828 Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  \/  ps )  \/  ch )  ->  ( ( (
 ph  \/  -.  ch )  \/  th )  ->  (
 ( ph  \/  ps )  \/  th ) ) )
 
Theorempm2.85 829 Theorem *2.85 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)
 |-  ( ( ( ph  \/  ps )  ->  ( ph  \/  ch ) ) 
 ->  ( ph  \/  ( ps  ->  ch ) ) )
 
Theorempm3.2ni 830 Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.)
 |- 
 -.  ph   &    |-  -.  ps   =>    |-  -.  ( ph  \/  ps )
 
Theoremorabs 831 Absorption of redundant internal disjunct. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 28-Feb-2014.)
 |-  ( ph  <->  ( ( ph  \/  ps )  /\  ph )
 )
 
Theoremoranabs 832 Absorb a disjunct into a conjunct. (Contributed by Roy F. Longton 23-Jun-2005.) (Proof shortened by Wolf Lammen, 10-Nov-2013.)
 |-  ( ( ( ph  \/  -.  ps )  /\  ps )  <->  ( ph  /\  ps ) )
 
Theorempm5.1 833 Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123. (Contributed by NM, 21-May-1994.)
 |-  ( ( ph  /\  ps )  ->  ( ph  <->  ps ) )
 
Theorempm5.21 834 Two propositions are equivalent if they are both false. Theorem *5.21 of [WhiteheadRussell] p. 124. (Contributed by NM, 21-May-1994.)
 |-  ( ( -.  ph  /\ 
 -.  ps )  ->  ( ph 
 <->  ps ) )
 
Theorempm3.43 835 Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  /\ 
 ch ) ) )
 
Theoremjcab 836 Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 27-Nov-2013.)
 |-  ( ( ph  ->  ( ps  /\  ch )
 ) 
 <->  ( ( ph  ->  ps )  /\  ( ph  ->  ch ) ) )
 
Theoremordi 837 Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 28-Nov-2013.)
 |-  ( ( ph  \/  ( ps  /\  ch )
 ) 
 <->  ( ( ph  \/  ps )  /\  ( ph  \/  ch ) ) )
 
Theoremordir 838 Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
 |-  ( ( ( ph  /\ 
 ps )  \/  ch ) 
 <->  ( ( ph  \/  ch )  /\  ( ps 
 \/  ch ) ) )
 
TheoremjcabOLD 839 Obsolete proof of jcab 836 as of 27-Nov-2013 (Contributed by NM, 3-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ph  ->  ( ps  /\  ch )
 ) 
 <->  ( ( ph  ->  ps )  /\  ( ph  ->  ch ) ) )
 
Theorempm3.43OLD 840 Obsolete proof of pm3.43 835 as of 27-Nov-2013 (Contributed by NM, 3-Jan-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  /\ 
 ch ) ) )
 
Theorempm4.76 841 Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  ch ) )  <->  ( ph  ->  ( ps  /\  ch )
 ) )
 
Theoremandi 842 Distributive law for conjunction. Theorem *4.4 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)
 |-  ( ( ph  /\  ( ps  \/  ch ) )  <-> 
 ( ( ph  /\  ps )  \/  ( ph  /\  ch ) ) )
 
Theoremandir 843 Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
 |-  ( ( ( ph  \/  ps )  /\  ch ) 
 <->  ( ( ph  /\  ch )  \/  ( ps  /\  ch ) ) )
 
Theoremorddi 844 Double distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
 |-  ( ( ( ph  /\ 
 ps )  \/  ( ch  /\  th ) )  <-> 
 ( ( ( ph  \/  ch )  /\  ( ph  \/  th ) ) 
 /\  ( ( ps 
 \/  ch )  /\  ( ps  \/  th ) ) ) )
 
Theoremanddi 845 Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
 |-  ( ( ( ph  \/  ps )  /\  ( ch  \/  th ) )  <-> 
 ( ( ( ph  /\ 
 ch )  \/  ( ph  /\  th ) )  \/  ( ( ps 
 /\  ch )  \/  ( ps  /\  th ) ) ) )
 
Theorempm4.39 846 Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  <->  ch )  /\  ( ps  <->  th ) )  ->  ( ( ph  \/  ps )  <->  ( ch  \/  th ) ) )
 
Theorempm4.38 847 Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  <->  ch )  /\  ( ps  <->  th ) )  ->  ( ( ph  /\  ps ) 
 <->  ( ch  /\  th ) ) )
 
Theorembi2anan9 848 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 31-Jul-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ta  <->  et ) )   =>    |-  ( ( ph  /\ 
 th )  ->  (
 ( ps  /\  ta ) 
 <->  ( ch  /\  et ) ) )
 
Theorembi2anan9r 849 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 19-Feb-1996.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ta  <->  et ) )   =>    |-  ( ( th  /\  ph )  ->  ( ( ps  /\  ta )  <->  ( ch  /\  et )
 ) )
 
Theorembi2bian9 850 Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ta  <->  et ) )   =>    |-  ( ( ph  /\ 
 th )  ->  (
 ( ps  <->  ta )  <->  ( ch  <->  et ) ) )
 
Theorempm4.72 851 Implication in terms of biconditional and disjunction. Theorem *4.72 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Jan-2013.)
 |-  ( ( ph  ->  ps )  <->  ( ps  <->  ( ph  \/  ps ) ) )
 
Theoremimimorb 852 Simplify an implication between implications. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
 |-  ( ( ( ps 
 ->  ch )  ->  ( ph  ->  ch ) )  <->  ( ph  ->  ( ps  \/  ch )
 ) )
 
Theorempm5.33 853 Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  /\  ( ps  ->  ch ) )  <->  ( ph  /\  (
 ( ph  /\  ps )  ->  ch ) ) )
 
Theorempm5.36 854 Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  /\  ( ph 
 <->  ps ) )  <->  ( ps  /\  ( ph  <->  ps ) ) )
 
Theorembianabs 855 Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007.)
 |-  ( ph  ->  ( ps 
 <->  ( ph  /\  ch ) ) )   =>    |-  ( ph  ->  ( ps  <->  ch ) )
 
Theoremoibabs 856 Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)
 |-  ( ( ( ph  \/  ps )  ->  ( ph 
 <->  ps ) )  <->  ( ph  <->  ps ) )
 
Theorempm3.24 857 Law of noncontradiction. Theorem *3.24 of [WhiteheadRussell] p. 111 (who call it the "law of contradiction"). (Contributed by NM, 16-Sep-1993.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
 |- 
 -.  ( ph  /\  -.  ph )
 
Theorempm2.26 858 Theorem *2.26 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
 |-  ( -.  ph  \/  ( ( ph  ->  ps )  ->  ps )
 )
 
Theorempm5.11 859 Theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ps )  \/  ( -.  ph  ->  ps ) )
 
Theorempm5.12 860 Theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ps )  \/  ( ph  ->  -.  ps ) )
 
Theorempm5.14 861 Theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ps )  \/  ( ps 
 ->  ch ) )
 
Theorempm5.13 862 Theorem *5.13 of [WhiteheadRussell] p. 123. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
 |-  ( ( ph  ->  ps )  \/  ( ps 
 ->  ph ) )
 
Theorempm5.17 863 Theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Jan-2013.)
 |-  ( ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps )
 ) 
 <->  ( ph  <->  -.  ps ) )
 
Theorempm5.15 864 Theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 15-Oct-2013.)
 |-  ( ( ph  <->  ps )  \/  ( ph 
 <->  -.  ps ) )
 
Theorempm5.16 865 Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 17-Oct-2013.)
 |- 
 -.  ( ( ph  <->  ps )  /\  ( ph  <->  -.  ps ) )
 
Theoremxor 866 Two ways to express "exclusive or." Theorem *5.22 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 22-Jan-2013.)
 |-  ( -.  ( ph  <->  ps ) 
 <->  ( ( ph  /\  -.  ps )  \/  ( ps 
 /\  -.  ph ) ) )
 
Theoremnbi2 867 Two ways to express "exclusive or." (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Jan-2013.)
 |-  ( -.  ( ph  <->  ps ) 
 <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps ) ) )
 
Theoremdfbi3 868 An alternate definition of the biconditional. Theorem *5.23 of [WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)
 |-  ( ( ph  <->  ps )  <->  ( ( ph  /\ 
 ps )  \/  ( -.  ph  /\  -.  ps ) ) )
 
Theorempm5.24 869 Theorem *5.24 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( (
 ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) )  <->  ( ( ph  /\ 
 -.  ps )  \/  ( ps  /\  -.  ph )
 ) )
 
Theoremxordi 870 Conjunction distributes over exclusive-or, using  -.  ( ph  <->  ps ) to express exclusive-or. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic. (Contributed by NM, 3-Oct-2008.)
 |-  ( ( ph  /\  -.  ( ps  <->  ch ) )  <->  -.  ( ( ph  /\ 
 ps )  <->  ( ph  /\  ch ) ) )
 
Theorembiort 871 A wff disjoined with truth is true. (Contributed by NM, 23-May-1999.)
 |-  ( ph  ->  ( ph 
 <->  ( ph  \/  ps ) ) )
 
Theorempm5.55 872 Theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 20-Jan-2013.)
 |-  ( ( ( ph  \/  ps )  <->  ph )  \/  (
 ( ph  \/  ps )  <->  ps ) )
 
1.3.7  Miscellaneous theorems of propositional calculus
 
Theorempm5.21nd 873 Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen, 4-Nov-2013.)
 |-  ( ( ph  /\  ps )  ->  th )   &    |-  ( ( ph  /\ 
 ch )  ->  th )   &    |-  ( th  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( ps  <->  ch ) )
 
Theorempm5.35 874 Theorem *5.35 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  <->  ch ) ) )
 
Theorempm5.54 875 Theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 7-Nov-2013.)
 |-  ( ( ( ph  /\ 
 ps )  <->  ph )  \/  (
 ( ph  /\  ps )  <->  ps ) )
 
Theorembaib 876 Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.)
 |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ps  ->  ( ph 
 <->  ch ) )
 
Theorembaibr 877 Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.)
 |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ps  ->  ( ch 
 <-> 
 ph ) )
 
Theoremrbaib 878 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ch  ->  ( ph 
 <->  ps ) )
 
Theoremrbaibr 879 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ch  ->  ( ps 
 <-> 
 ph ) )
 
Theorembaibd 880 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  ( ph  ->  ( ps 
 <->  ( ch  /\  th ) ) )   =>    |-  ( ( ph  /\ 
 ch )  ->  ( ps 
 <-> 
 th ) )
 
Theoremrbaibd 881 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  ( ph  ->  ( ps 
 <->  ( ch  /\  th ) ) )   =>    |-  ( ( ph  /\ 
 th )  ->  ( ps 
 <->  ch ) )
 
Theorempm5.44 882 Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ps )  ->  ( ( ph  ->  ch )  <->  ( ph  ->  ( ps  /\  ch )
 ) ) )
 
Theorempm5.6 883 Conjunction in antecedent versus disjunction in consequent. Theorem *5.6 of [WhiteheadRussell] p. 125. (Contributed by NM, 8-Jun-1994.)
 |-  ( ( ( ph  /\ 
 -.  ps )  ->  ch )  <->  (
 ph  ->  ( ps  \/  ch ) ) )
 
Theoremorcanai 884 Change disjunction in consequent to conjunction in antecedent. (Contributed by NM, 8-Jun-1994.)
 |-  ( ph  ->  ( ps  \/  ch ) )   =>    |-  ( ( ph  /\  -.  ps )  ->  ch )
 
Theoremintnan 885 Introduction of conjunct inside of a contradiction. (Contributed by NM, 16-Sep-1993.)
 |- 
 -.  ph   =>    |- 
 -.  ( ps  /\  ph )
 
Theoremintnanr 886 Introduction of conjunct inside of a contradiction. (Contributed by NM, 3-Apr-1995.)
 |- 
 -.  ph   =>    |- 
 -.  ( ph  /\  ps )
 
Theoremintnand 887 Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  -.  ( ch  /\  ps ) )
 
Theoremintnanrd 888 Introduction of conjunct inside of a contradiction. (Contributed by NM, 10-Jul-2005.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  -.  ( ps  /\  ch ) )
 
Theoremmpbiran 889 Detach truth from conjunction in biconditional. (Contributed by NM, 27-Feb-1996.)
 |- 
 ps   &    |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ph  <->  ch )
 
Theoremmpbiran2 890 Detach truth from conjunction in biconditional. (Contributed by NM, 22-Feb-1996.)
 |- 
 ch   &    |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ph  <->  ps )
 
Theoremmpbir2an 891 Detach a conjunction of truths in a biconditional. (Contributed by NM, 10-May-2005.)
 |- 
 ps   &    |- 
 ch   &    |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ph
 
Theoremmpbi2and 892 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 893 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 )
 
Theorempm5.62 894 Theorem *5.62 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 21-Jun-2005.)
 |-  ( ( ( ph  /\ 
 ps )  \/  -.  ps )  <->  ( ph  \/  -. 
 ps ) )
 
Theorempm5.63 895 Theorem *5.63 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 25-Dec-2012.)
 |-  ( ( ph  \/  ps )  <->  ( ph  \/  ( -.  ph  /\  ps )
 ) )
 
Theorembianfi 896 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 897 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 ) ) )
 
Theorempm4.43 898 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 )
 ) )
 
Theorempm4.82 899 Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  -.  ps )
 ) 
 <->  -.  ph )
 
Theorempm4.83 900 Theorem *4.83 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  ->  ps )  /\  ( -.  ph  ->  ps )
 ) 
 <->  ps )
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