P. 81 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8001-8100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	addsubeq4 8001	Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) = ( C + D ) <-> ( C - A ) = ( B - D ) ) )
Theorem	pncan3oi 8002	Subtraction and addition of equals. Almost but not exactly the same as pncan3i 8063 and pncan 7992, this order happens often when applying "operations to both sides" so create a theorem specifically for it. A deduction version of this is available as pncand 8098. (Contributed by David A. Wheeler, 11-Oct-2018.)
|- A e. CC   &    |- B e. CC   =>    |- ( ( A + B ) - B ) = A
Theorem	mvrraddi 8003	Move RHS right addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
|- B e. CC   &    |- C e. CC   &    |- A = ( B + C )   =>    |- ( A - C ) = B
Theorem	mvlladdi 8004	Move LHS left addition to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
|- A e. CC   &    |- B e. CC   &    |- ( A + B ) = C   =>    |- B = ( C - A )
Theorem	subid 8005	Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( A e. CC -> ( A - A ) = 0 )
Theorem	subid1 8006	Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( A e. CC -> ( A - 0 ) = A )
Theorem	npncan 8007	Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) + ( B - C ) ) = ( A - C ) )
Theorem	nppcan 8008	Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A - B ) + C ) + B ) = ( A + C ) )
Theorem	nnpcan 8009	Cancellation law for subtraction: ((a-b)-c)+b = a-c holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 24-Mar-2018.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A - B ) - C ) + B ) = ( A - C ) )
Theorem	nppcan3 8010	Cancellation law for subtraction. (Contributed by Mario Carneiro, 14-Sep-2015.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) + ( C + B ) ) = ( A + C ) )
Theorem	subcan2 8011	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 ) )
Theorem	subeq0 8012	If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) = 0 <-> A = B ) )
Theorem	npncan2 8013	Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) + ( B - A ) ) = 0 )
Theorem	subsub2 8014	Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B - C ) ) = ( A + ( C - B ) ) )
Theorem	nncan 8015	Cancellation law for subtraction. (Contributed by NM, 21-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( ( A e. CC /\ B e. CC ) -> ( A - ( A - B ) ) = B )
Theorem	subsub 8016	Law for double subtraction. (Contributed by NM, 13-May-2004.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B - C ) ) = ( ( A - B ) + C ) )
Theorem	nppcan2 8017	Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - ( B + C ) ) + C ) = ( A - B ) )
Theorem	subsub3 8018	Law for double subtraction. (Contributed by NM, 27-Jul-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - ( B - C ) ) = ( ( A + C ) - B ) )
Theorem	subsub4 8019	Law for double subtraction. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) - C ) = ( A - ( B + C ) ) )
Theorem	sub32 8020	Swap the second and third terms in a double subtraction. (Contributed by NM, 19-Aug-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) - C ) = ( ( A - C ) - B ) )
Theorem	nnncan 8021	Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - ( B - C ) ) - C ) = ( A - B ) )
Theorem	nnncan1 8022	Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) - ( A - C ) ) = ( C - B ) )
Theorem	nnncan2 8023	Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - C ) - ( B - C ) ) = ( A - B ) )
Theorem	npncan3 8024	Cancellation law for subtraction. (Contributed by Scott Fenton, 23-Jun-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) + ( C - A ) ) = ( C - B ) )
Theorem	pnpcan 8025	Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( A + C ) ) = ( B - C ) )
Theorem	pnpcan2 8026	Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) - ( B + C ) ) = ( A - B ) )
Theorem	pnncan 8027	Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( A - C ) ) = ( B + C ) )
Theorem	ppncan 8028	Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + ( C - B ) ) = ( A + C ) )
Theorem	addsub4 8029	Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) - ( C + D ) ) = ( ( A - C ) + ( B - D ) ) )
Theorem	subadd4 8030	Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 24-Aug-2006.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A - B ) - ( C - D ) ) = ( ( A + D ) - ( B + C ) ) )
Theorem	sub4 8031	Rearrangement of 4 terms in a subtraction. (Contributed by NM, 23-Nov-2007.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A - B ) - ( C - D ) ) = ( ( A - C ) - ( B - D ) ) )
Theorem	neg0 8032	Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
|- -u 0 = 0
Theorem	negid 8033	Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)
|- ( A e. CC -> ( A + -u A ) = 0 )
Theorem	negsub 8034	Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
Theorem	subneg 8035	Relationship between subtraction and negative. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> ( A - -u B ) = ( A + B ) )
Theorem	negneg 8036	A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by NM, 12-Jan-2002.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( A e. CC -> -u -u A = A )
Theorem	neg11 8037	Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> ( -u A = -u B <-> A = B ) )
Theorem	negcon1 8038	Negative contraposition law. (Contributed by NM, 9-May-2004.)
|- ( ( A e. CC /\ B e. CC ) -> ( -u A = B <-> -u B = A ) )
Theorem	negcon2 8039	Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
|- ( ( A e. CC /\ B e. CC ) -> ( A = -u B <-> B = -u A ) )
Theorem	negeq0 8040	A number is zero iff its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( A e. CC -> ( A = 0 <-> -u A = 0 ) )
Theorem	subcan 8041	Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = ( A - C ) <-> B = C ) )
Theorem	negsubdi 8042	Distribution of negative over subtraction. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( -u A + B ) )
Theorem	negdi 8043	Distribution of negative over addition. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> -u ( A + B ) = ( -u A + -u B ) )
Theorem	negdi2 8044	Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
|- ( ( A e. CC /\ B e. CC ) -> -u ( A + B ) = ( -u A - B ) )
Theorem	negsubdi2 8045	Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)
|- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( B - A ) )
Theorem	neg2sub 8046	Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( -u A - -u B ) = ( B - A ) )
Theorem	renegcl 8047	Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.)
|- ( A e. RR -> -u A e. RR )
Theorem	renegcli 8048	Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 8047 is deprecated, but is retained for its demonstration value.) (Contributed by NM, 17-Jan-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- A e. RR   =>    |- -u A e. RR
Theorem	resubcli 8049	Closure law for subtraction of reals. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.)
|- A e. RR   &    |- B e. RR   =>    |- ( A - B ) e. RR
Theorem	resubcl 8050	Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)
|- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR )
Theorem	negreb 8051	The negative of a real is real. (Contributed by NM, 11-Aug-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
|- ( A e. CC -> ( -u A e. RR <-> A e. RR ) )
Theorem	peano2cnm 8052	"Reverse" second Peano postulate analog for complex numbers: A complex number minus 1 is a complex number. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
|- ( N e. CC -> ( N - 1 ) e. CC )
Theorem	peano2rem 8053	"Reverse" second Peano postulate analog for reals. (Contributed by NM, 6-Feb-2007.)
|- ( N e. RR -> ( N - 1 ) e. RR )
Theorem	negcli 8054	Closure law for negative. (Contributed by NM, 26-Nov-1994.)
|- A e. CC   =>    |- -u A e. CC
Theorem	negidi 8055	Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)
|- A e. CC   =>    |- ( A + -u A ) = 0
Theorem	negnegi 8056	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
Theorem	subidi 8057	Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)
|- A e. CC   =>    |- ( A - A ) = 0
Theorem	subid1i 8058	Identity law for subtraction. (Contributed by NM, 29-May-1999.)
|- A e. CC   =>    |- ( A - 0 ) = A
Theorem	negne0bi 8059	A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)
|- A e. CC   =>    |- ( A =/= 0 <-> -u A =/= 0 )
Theorem	negrebi 8060	The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
|- A e. CC   =>    |- ( -u A e. RR <-> A e. RR )
Theorem	negne0i 8061	The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)
|- A e. CC   &    |- A =/= 0   =>    |- -u A =/= 0
Theorem	subcli 8062	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
Theorem	pncan3i 8063	Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)
|- A e. CC   &    |- B e. CC   =>    |- ( A + ( B - A ) ) = B
Theorem	negsubi 8064	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 )
Theorem	subnegi 8065	Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)
|- A e. CC   &    |- B e. CC   =>    |- ( A - -u B ) = ( A + B )
Theorem	subeq0i 8066	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 )
Theorem	neg11i 8067	Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( -u A = -u B <-> A = B )
Theorem	negcon1i 8068	Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( -u A = B <-> -u B = A )
Theorem	negcon2i 8069	Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( A = -u B <-> B = -u A )
Theorem	negdii 8070	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 )
Theorem	negsubdii 8071	Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)
|- A e. CC   &    |- B e. CC   =>    |- -u ( A - B ) = ( -u A + B )
Theorem	negsubdi2i 8072	Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)
|- A e. CC   &    |- B e. CC   =>    |- -u ( A - B ) = ( B - A )
Theorem	subaddi 8073	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 )
Theorem	subadd2i 8074	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 )
Theorem	subaddrii 8075	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
Theorem	subsub23i 8076	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 )
Theorem	addsubassi 8077	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 ) )
Theorem	addsubi 8078	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 )
Theorem	subcani 8079	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 )
Theorem	subcan2i 8080	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 )
Theorem	pnncani 8081	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 )
Theorem	addsub4i 8082	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 ) )
Theorem	0reALT 8083	Alternate proof of 0re 7790. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
|- 0 e. RR
Theorem	negcld 8084	Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> -u A e. CC )
Theorem	subidd 8085	Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A - A ) = 0 )
Theorem	subid1d 8086	Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A - 0 ) = A )
Theorem	negidd 8087	Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A + -u A ) = 0 )
Theorem	negnegd 8088	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 )
Theorem	negeq0d 8089	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 ) )
Theorem	negne0bd 8090	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 ) )
Theorem	negcon1d 8091	Contraposition law for unary minus. Deduction form of negcon1 8038. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( -u A = B <-> -u B = A ) )
Theorem	negcon1ad 8092	Contraposition law for unary minus. One-way deduction form of negcon1 8038. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> -u A = B )   =>    |- ( ph -> -u B = A )
Theorem	neg11ad 8093	The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 8037. Generalization of neg11d 8109. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( -u A = -u B <-> A = B ) )
Theorem	negned 8094	If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 8109. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A =/= B )   =>    |- ( ph -> -u A =/= -u B )
Theorem	negne0d 8095	The negative of a nonzero number is nonzero. See also negap0d 8417 which is similar but for apart from zero rather than not equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> A =/= 0 )   =>    |- ( ph -> -u A =/= 0 )
Theorem	negrebd 8096	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 )
Theorem	subcld 8097	Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( A - B ) e. CC )
Theorem	pncand 8098	Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( ( A + B ) - B ) = A )
Theorem	pncan2d 8099	Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( ( A + B ) - A ) = B )
Theorem	pncan3d 8100	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 )