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Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnegcli 7901 Closure law for negative. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   =>    |-  -u A  e.  CC
 
Theoremnegidi 7902 Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   =>    |-  ( A  +  -u A )  =  0
 
Theoremnegnegi 7903 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by NM, 8-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  A  e.  CC   =>    |-  -u -u A  =  A
 
Theoremsubidi 7904 Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   =>    |-  ( A  -  A )  =  0
 
Theoremsubid1i 7905 Identity law for subtraction. (Contributed by NM, 29-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  -  0 )  =  A
 
Theoremnegne0bi 7906 A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0 
 <->  -u A  =/=  0
 )
 
Theoremnegrebi 7907 The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( -u A  e.  RR  <->  A  e.  RR )
 
Theoremnegne0i 7908 The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  -u A  =/=  0
 
Theoremsubcli 7909 Closure law for subtraction. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  -  B )  e.  CC
 
Theorempncan3i 7910 Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  +  ( B  -  A ) )  =  B
 
Theoremnegsubi 7911 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  +  -u B )  =  ( A  -  B )
 
Theoremsubnegi 7912 Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  -  -u B )  =  ( A  +  B )
 
Theoremsubeq0i 7913 If the difference between two numbers is zero, they are equal. (Contributed by NM, 8-May-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( A  -  B )  =  0  <->  A  =  B )
 
Theoremneg11i 7914 Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( -u A  =  -u B 
 <->  A  =  B )
 
Theoremnegcon1i 7915 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( -u A  =  B  <->  -u B  =  A )
 
Theoremnegcon2i 7916 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  =  -u B 
 <->  B  =  -u A )
 
Theoremnegdii 7917 Distribution of negative over addition. (Contributed by NM, 28-Jul-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  -u ( A  +  B )  =  ( -u A  +  -u B )
 
Theoremnegsubdii 7918 Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  -u ( A  -  B )  =  ( -u A  +  B )
 
Theoremnegsubdi2i 7919 Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  -u ( A  -  B )  =  ( B  -  A )
 
Theoremsubaddi 7920 Relationship between subtraction and addition. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  -  B )  =  C  <->  ( B  +  C )  =  A )
 
Theoremsubadd2i 7921 Relationship between subtraction and addition. (Contributed by NM, 15-Dec-2006.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  -  B )  =  C  <->  ( C  +  B )  =  A )
 
Theoremsubaddrii 7922 Relationship between subtraction and addition. (Contributed by NM, 16-Dec-2006.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  ( B  +  C )  =  A   =>    |-  ( A  -  B )  =  C
 
Theoremsubsub23i 7923 Swap subtrahend and result of subtraction. (Contributed by NM, 7-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  -  B )  =  C  <->  ( A  -  C )  =  B )
 
Theoremaddsubassi 7924 Associative-type law for subtraction and addition. (Contributed by NM, 16-Sep-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  -  C )  =  ( A  +  ( B  -  C ) )
 
Theoremaddsubi 7925 Law for subtraction and addition. (Contributed by NM, 6-Aug-2003.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  -  C )  =  ( ( A  -  C )  +  B )
 
Theoremsubcani 7926 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  -  B )  =  ( A  -  C )  <->  B  =  C )
 
Theoremsubcan2i 7927 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  -  C )  =  ( B  -  C )  <->  A  =  B )
 
Theorempnncani 7928 Cancellation law for mixed addition and subtraction. (Contributed by NM, 14-Jan-2006.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  -  ( A  -  C ) )  =  ( B  +  C )
 
Theoremaddsub4i 7929 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 17-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   =>    |-  ( ( A  +  B )  -  ( C  +  D )
 )  =  ( ( A  -  C )  +  ( B  -  D ) )
 
Theorem0reALT 7930 Alternate proof of 0re 7638. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  0  e.  RR
 
Theoremnegcld 7931 Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  -u A  e.  CC )
 
Theoremsubidd 7932 Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  -  A )  =  0 )
 
Theoremsubid1d 7933 Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  -  0 )  =  A )
 
Theoremnegidd 7934 Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  +  -u A )  =  0 )
 
Theoremnegnegd 7935 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  -u -u A  =  A )
 
Theoremnegeq0d 7936 A number is zero iff its negative is zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  =  0  <->  -u A  =  0 ) )
 
Theoremnegne0bd 7937 A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  =/=  0  <->  -u A  =/=  0
 ) )
 
Theoremnegcon1d 7938 Contraposition law for unary minus. Deduction form of negcon1 7885. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 -u A  =  B  <->  -u B  =  A )
 )
 
Theoremnegcon1ad 7939 Contraposition law for unary minus. One-way deduction form of negcon1 7885. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  -u A  =  B )   =>    |-  ( ph  ->  -u B  =  A )
 
Theoremneg11ad 7940 The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 7884. Generalization of neg11d 7956. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 -u A  =  -u B 
 <->  A  =  B ) )
 
Theoremnegned 7941 If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 7956. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  -u A  =/=  -u B )
 
Theoremnegne0d 7942 The negative of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  -u A  =/=  0 )
 
Theoremnegrebd 7943 The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  -u A  e.  RR )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremsubcld 7944 Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  -  B )  e.  CC )
 
Theorempncand 7945 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  +  B )  -  B )  =  A )
 
Theorempncan2d 7946 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  +  B )  -  A )  =  B )
 
Theorempncan3d 7947 Subtraction and addition of equals. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  +  ( B  -  A ) )  =  B )
 
Theoremnpcand 7948 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  -  B )  +  B )  =  A )
 
Theoremnncand 7949 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  -  ( A  -  B ) )  =  B )
 
Theoremnegsubd 7950 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  +  -u B )  =  ( A  -  B ) )
 
Theoremsubnegd 7951 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  -  -u B )  =  ( A  +  B ) )
 
Theoremsubeq0d 7952 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A  -  B )  =  0
 )   =>    |-  ( ph  ->  A  =  B )
 
Theoremsubne0d 7953 Two unequal numbers have nonzero difference. See also subap0d 8271 which is the same thing for apartness rather than negated equality. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ( A  -  B )  =/=  0 )
 
Theoremsubeq0ad 7954 The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 7859. Generalization of subeq0d 7952. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  -  B )  =  0  <->  A  =  B ) )
 
Theoremsubne0ad 7955 If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d 7953. Contrapositive of subeq0bd 8008. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A  -  B )  =/=  0
 )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremneg11d 7956 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  -u A  =  -u B )   =>    |-  ( ph  ->  A  =  B )
 
Theoremnegdid 7957 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  -u ( A  +  B )  =  ( -u A  +  -u B ) )
 
Theoremnegdi2d 7958 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  -u ( A  +  B )  =  ( -u A  -  B ) )
 
Theoremnegsubdid 7959 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  -u ( A  -  B )  =  ( -u A  +  B ) )
 
Theoremnegsubdi2d 7960 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  -u ( A  -  B )  =  ( B  -  A ) )
 
Theoremneg2subd 7961 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 -u A  -  -u B )  =  ( B  -  A ) )
 
Theoremsubaddd 7962 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  =  C  <->  ( B  +  C )  =  A ) )
 
Theoremsubadd2d 7963 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  =  C  <->  ( C  +  B )  =  A ) )
 
Theoremaddsubassd 7964 Associative-type law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  -  C )  =  ( A  +  ( B  -  C ) ) )
 
Theoremaddsubd 7965 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  -  C )  =  ( ( A  -  C )  +  B ) )
 
Theoremsubadd23d 7966 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  +  C )  =  ( A  +  ( C  -  B ) ) )
 
Theoremaddsub12d 7967 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  +  ( B  -  C ) )  =  ( B  +  ( A  -  C ) ) )
 
Theoremnpncand 7968 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  +  ( B  -  C ) )  =  ( A  -  C ) )
 
Theoremnppcand 7969 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( ( A  -  B )  +  C )  +  B )  =  ( A  +  C ) )
 
Theoremnppcan2d 7970 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  ( B  +  C )
 )  +  C )  =  ( A  -  B ) )
 
Theoremnppcan3d 7971 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  +  ( C  +  B ) )  =  ( A  +  C ) )
 
Theoremsubsubd 7972 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  -  ( B  -  C ) )  =  ( ( A  -  B )  +  C ) )
 
Theoremsubsub2d 7973 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  -  ( B  -  C ) )  =  ( A  +  ( C  -  B ) ) )
 
Theoremsubsub3d 7974 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  -  ( B  -  C ) )  =  ( ( A  +  C )  -  B ) )
 
Theoremsubsub4d 7975 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  -  C )  =  ( A  -  ( B  +  C )
 ) )
 
Theoremsub32d 7976 Swap the second and third terms in a double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  -  C )  =  ( ( A  -  C )  -  B ) )
 
Theoremnnncand 7977 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  ( B  -  C ) )  -  C )  =  ( A  -  B ) )
 
Theoremnnncan1d 7978 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  -  ( A  -  C ) )  =  ( C  -  B ) )
 
Theoremnnncan2d 7979 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  C )  -  ( B  -  C ) )  =  ( A  -  B ) )
 
Theoremnpncan3d 7980 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  +  ( C  -  A ) )  =  ( C  -  B ) )
 
Theorempnpcand 7981 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  -  ( A  +  C ) )  =  ( B  -  C ) )
 
Theorempnpcan2d 7982 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  C )  -  ( B  +  C ) )  =  ( A  -  B ) )
 
Theorempnncand 7983 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  -  ( A  -  C ) )  =  ( B  +  C ) )
 
Theoremppncand 7984 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  +  ( C  -  B ) )  =  ( A  +  C ) )
 
Theoremsubcand 7985 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  ( A  -  B )  =  ( A  -  C ) )   =>    |-  ( ph  ->  B  =  C )
 
Theoremsubcan2d 7986 Cancellation law for subtraction. (Contributed by Mario Carneiro, 22-Sep-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  ( A  -  C )  =  ( B  -  C ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremsubcanad 7987 Cancellation law for subtraction. Deduction form of subcan 7888. Generalization of subcand 7985. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  =  ( A  -  C )  <->  B  =  C ) )
 
Theoremsubneintrd 7988 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcand 7985. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  ( A  -  B )  =/=  ( A  -  C ) )
 
Theoremsubcan2ad 7989 Cancellation law for subtraction. Deduction form of subcan2 7858. Generalization of subcan2d 7986. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  C )  =  ( B  -  C )  <->  A  =  B ) )
 
Theoremsubneintr2d 7990 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcan2d 7986. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ( A  -  C )  =/=  ( B  -  C ) )
 
Theoremaddsub4d 7991 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  -  ( C  +  D ) )  =  ( ( A  -  C )  +  ( B  -  D ) ) )
 
Theoremsubadd4d 7992 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  -  ( C  -  D ) )  =  ( ( A  +  D )  -  ( B  +  C )
 ) )
 
Theoremsub4d 7993 Rearrangement of 4 terms in a subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  -  ( C  -  D ) )  =  ( ( A  -  C )  -  ( B  -  D ) ) )
 
Theorem2addsubd 7994 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( ( A  +  B )  +  C )  -  D )  =  ( ( ( A  +  C )  -  D )  +  B ) )
 
Theoremaddsubeq4d 7995 Relation between sums and differences. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  =  ( C  +  D )  <->  ( C  -  A )  =  ( B  -  D ) ) )
 
Theoremmvlraddd 7996 Move LHS right addition to RHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A  +  B )  =  C )   =>    |-  ( ph  ->  A  =  ( C  -  B ) )
 
Theoremmvrraddd 7997 Move RHS right addition to LHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =  ( B  +  C ) )   =>    |-  ( ph  ->  ( A  -  C )  =  B )
 
Theoremsubaddeqd 7998 Transfer two terms of a subtraction to an addition in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  ( A  +  B )  =  ( C  +  D ) )   =>    |-  ( ph  ->  ( A  -  D )  =  ( C  -  B ) )
 
Theoremaddlsub 7999 Left-subtraction: Subtraction of the left summand from the result of an addition. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  =  C  <->  A  =  ( C  -  B ) ) )
 
Theoremaddrsub 8000 Right-subtraction: Subtraction of the right summand from the result of an addition. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  =  C  <->  B  =  ( C  -  A ) ) )
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