P. 84 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8301-8400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	negcon1i 8301	Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( -u A = B <-> -u B = A )
Theorem	negcon2i 8302	Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
|- A e. CC   &    |- B e. CC   =>    |- ( A = -u B <-> B = -u A )
Theorem	negdii 8303	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 8304	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 8305	Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)
|- A e. CC   &    |- B e. CC   =>    |- -u ( A - B ) = ( B - A )
Theorem	subaddi 8306	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 8307	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 8308	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 8309	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 8310	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 8311	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 8312	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 8313	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 8314	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 8315	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 8316	Alternate proof of 0re 8019. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
|- 0 e. RR
Theorem	negcld 8317	Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> -u A e. CC )
Theorem	subidd 8318	Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A - A ) = 0 )
Theorem	subid1d 8319	Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A - 0 ) = A )
Theorem	negidd 8320	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 8321	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 8322	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 8323	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 8324	Contraposition law for unary minus. Deduction form of negcon1 8271. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( -u A = B <-> -u B = A ) )
Theorem	negcon1ad 8325	Contraposition law for unary minus. One-way deduction form of negcon1 8271. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> -u A = B )   =>    |- ( ph -> -u B = A )
Theorem	neg11ad 8326	The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 8270. Generalization of neg11d 8342. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( -u A = -u B <-> A = B ) )
Theorem	negned 8327	If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 8342. (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 8328	The negative of a nonzero number is nonzero. See also negap0d 8650 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 8329	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 8330	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 8331	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 8332	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 8333	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 )
Theorem	npcand 8334	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	nncand 8335	Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( A - ( A - B ) ) = B )
Theorem	negsubd 8336	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 ) )
Theorem	subnegd 8337	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 ) )
Theorem	subeq0d 8338	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 )
Theorem	subne0d 8339	Two unequal numbers have nonzero difference. See also subap0d 8663 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 )
Theorem	subeq0ad 8340	The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 8245. Generalization of subeq0d 8338. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( ( A - B ) = 0 <-> A = B ) )
Theorem	subne0ad 8341	If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d 8339. Contrapositive of subeq0bd 8398. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> ( A - B ) =/= 0 )   =>    |- ( ph -> A =/= B )
Theorem	neg11d 8342	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 )
Theorem	negdid 8343	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 ) )
Theorem	negdi2d 8344	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 ) )
Theorem	negsubdid 8345	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 ) )
Theorem	negsubdi2d 8346	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 ) )
Theorem	neg2subd 8347	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 ) )
Theorem	subaddd 8348	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 ) )
Theorem	subadd2d 8349	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 ) )
Theorem	addsubassd 8350	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 ) ) )
Theorem	addsubd 8351	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 ) )
Theorem	subadd23d 8352	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 ) ) )
Theorem	addsub12d 8353	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 ) ) )
Theorem	npncand 8354	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 ) )
Theorem	nppcand 8355	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 ) )
Theorem	nppcan2d 8356	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 ) )
Theorem	nppcan3d 8357	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 ) )
Theorem	subsubd 8358	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 ) )
Theorem	subsub2d 8359	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 ) ) )
Theorem	subsub3d 8360	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 ) )
Theorem	subsub4d 8361	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 ) ) )
Theorem	sub32d 8362	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 ) )
Theorem	nnncand 8363	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 ) )
Theorem	nnncan1d 8364	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 ) )
Theorem	nnncan2d 8365	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 ) )
Theorem	npncan3d 8366	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 ) )
Theorem	pnpcand 8367	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 ) )
Theorem	pnpcan2d 8368	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 ) )
Theorem	pnncand 8369	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 ) )
Theorem	ppncand 8370	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 ) )
Theorem	subcand 8371	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 )
Theorem	subcan2d 8372	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 )
Theorem	subcanad 8373	Cancellation law for subtraction. Deduction form of subcan 8274. Generalization of subcand 8371. (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 ) )
Theorem	subneintrd 8374	Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcand 8371. (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 ) )
Theorem	subcan2ad 8375	Cancellation law for subtraction. Deduction form of subcan2 8244. Generalization of subcan2d 8372. (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 ) )
Theorem	subneintr2d 8376	Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcan2d 8372. (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 ) )
Theorem	addsub4d 8377	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 ) ) )
Theorem	subadd4d 8378	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 ) ) )
Theorem	sub4d 8379	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 ) ) )
Theorem	2addsubd 8380	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 ) )
Theorem	addsubeq4d 8381	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 ) ) )
Theorem	subeqxfrd 8382	Transfer two terms of a subtraction 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 - C ) = ( B - D ) )
Theorem	mvlraddd 8383	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 ) )
Theorem	mvlladdd 8384	Move LHS left addition to RHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> ( A + B ) = C )   =>    |- ( ph -> B = ( C - A ) )
Theorem	mvrraddd 8385	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 )
Theorem	mvrladdd 8386	Move RHS left addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
|- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> A = ( B + C ) )   =>    |- ( ph -> ( A - B ) = C )
Theorem	assraddsubd 8387	Associate RHS addition-subtraction. (Contributed by David A. Wheeler, 15-Oct-2018.)
|- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   &    |- ( ph -> A = ( ( B + C ) - D ) )   =>    |- ( ph -> A = ( B + ( C - D ) ) )
Theorem	subaddeqd 8388	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 ) )
Theorem	addlsub 8389	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 ) ) )
Theorem	addrsub 8390	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 ) ) )
Theorem	subexsub 8391	A subtraction law: Exchanging the subtrahend and the result of the subtraction. (Contributed by BJ, 6-Jun-2019.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( A = ( C - B ) <-> B = ( C - A ) ) )
Theorem	addid0 8392	If adding a number to a another number yields the other number, the added number must be 0. This shows that 0 is the unique (right) identity of the complex numbers. (Contributed by AV, 17-Jan-2021.)
|- ( ( X e. CC /\ Y e. CC ) -> ( ( X + Y ) = X <-> Y = 0 ) )
Theorem	addn0nid 8393	Adding a nonzero number to a complex number does not yield the complex number. (Contributed by AV, 17-Jan-2021.)
|- ( ( X e. CC /\ Y e. CC /\ Y =/= 0 ) -> ( X + Y ) =/= X )
Theorem	pnpncand 8394	Addition/subtraction cancellation law. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A + ( B - C ) ) + ( C - B ) ) = A )
Theorem	subeqrev 8395	Reverse the order of subtraction in an equality. (Contributed by Scott Fenton, 8-Jul-2013.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A - B ) = ( C - D ) <-> ( B - A ) = ( D - C ) ) )
Theorem	pncan1 8396	Cancellation law for addition and subtraction with 1. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
|- ( A e. CC -> ( ( A + 1 ) - 1 ) = A )
Theorem	npcan1 8397	Cancellation law for subtraction and addition with 1. (Contributed by Alexander van der Vekens, 5-Oct-2018.)
|- ( A e. CC -> ( ( A - 1 ) + 1 ) = A )
Theorem	subeq0bd 8398	If two complex numbers are equal, their difference is zero. Consequence of subeq0ad 8340. Converse of subeq0d 8338. Contrapositive of subne0ad 8341. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> A = B )   =>    |- ( ph -> ( A - B ) = 0 )
Theorem	renegcld 8399	Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> -u A e. RR )
Theorem	resubcld 8400	Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A - B ) e. RR )