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Theorem List for Metamath Proof Explorer - 1301-1400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nanbi1 1301	Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
|- ( ( ph <-> ps ) -> ( ( ph -/\ ch ) <-> ( ps -/\ ch ) ) )
Theorem	nanbi2 1302	Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
|- ( ( ph <-> ps ) -> ( ( ch -/\ ph ) <-> ( ch -/\ ps ) ) )
Theorem	nanbi12 1303	Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)
|- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ( ph -/\ ch ) <-> ( ps -/\ th ) ) )
Theorem	nanbi1i 1304	Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
|- ( ph <-> ps )   =>    |- ( ( ph -/\ ch ) <-> ( ps -/\ ch ) )
Theorem	nanbi2i 1305	Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
|- ( ph <-> ps )   =>    |- ( ( ch -/\ ph ) <-> ( ch -/\ ps ) )
Theorem	nanbi12i 1306	Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   =>    |- ( ( ph -/\ ch ) <-> ( ps -/\ th ) )
Theorem	nanbi1d 1307	Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( ps -/\ th ) <-> ( ch -/\ th ) ) )
Theorem	nanbi2d 1308	Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( th -/\ ps ) <-> ( th -/\ ch ) ) )
Theorem	nanbi12d 1309	Join two logical equivalences with anti-conjunction. (Contributed by Scott Fenton, 2-Jan-2018.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   =>    |- ( ph -> ( ( ps -/\ th ) <-> ( ch -/\ ta ) ) )
1.2.10  Logical 'xor'
Syntax	wxo 1310	Extend wff definition to include exclusive disjunction ('xor').
wff ( ph \/_ ps )
Definition	df-xor 1311	Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. After we define the constant true T. (df-tru 1325) and the constant false F. (df-fal 1326), we will be able to prove these truth table values: ( ( T. \/_ T. ) <-> F. ) (truxortru 1364), ( ( T. \/_ F. ) <-> T. ) (truxorfal 1365), ( ( F. \/_ T. ) <-> T. ) (falxortru 1366), and ( ( F. \/_ F. ) <-> F. ) (falxorfal 1367). Contrast with /\ (df-an 361), \/ (df-or 360), -> (wi 4), and -/\ (df-nan 1294) . (Contributed by FL, 22-Nov-2010.)
|- ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) )
Theorem	xnor 1312	Two ways to write XNOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ( ph <-> ps ) <-> -. ( ph \/_ ps ) )
Theorem	xorcom 1313	\/_ is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ( ph \/_ ps ) <-> ( ps \/_ ph ) )
Theorem	xorass 1314	\/_ is associative. (Contributed by FL, 22-Nov-2010.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- ( ( ( ph \/_ ps ) \/_ ch ) <-> ( ph \/_ ( ps \/_ ch ) ) )
Theorem	excxor 1315	This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.)
|- ( ( ph \/_ ps ) <-> ( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) )
Theorem	xor2 1316	Two ways to express "exclusive or." (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
Theorem	xorneg1 1317	\/_ is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ( -. ph \/_ ps ) <-> -. ( ph \/_ ps ) )
Theorem	xorneg2 1318	\/_ is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ( ph \/_ -. ps ) <-> -. ( ph \/_ ps ) )
Theorem	xorneg 1319	\/_ is unchanged under negation of both arguments. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ( -. ph \/_ -. ps ) <-> ( ph \/_ ps ) )
Theorem	xorbi12i 1320	Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   =>    |- ( ( ph \/_ ch ) <-> ( ps \/_ th ) )
Theorem	xorbi12d 1321	Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   =>    |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ ta ) ) )
1.2.11  True and false constants
Syntax	wtru 1322	T. is a wff.
wff T.
Syntax	wfal 1323	F. is a wff.
wff F.
Theorem	trujust 1324	Soundness justification theorem for df-tru 1325. (Contributed by Mario Carneiro, 17-Nov-2013.)
|- ( ( ph <-> ph ) <-> ( ps <-> ps ) )
Definition	df-tru 1325	Definition of T., a tautology. T. is a constant true. In this definition biid 228 is used as an antecedent, however, any true wff, such as an axiom, can be used in its place. (Contributed by Anthony Hart, 13-Oct-2010.)
|- ( T. <-> ( ph <-> ph ) )
Definition	df-fal 1326	Definition of F., a contradiction. F. is a constant false. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( F. <-> -. T. )
Theorem	tru 1327	T. is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
|- T.
Theorem	fal 1328	F. is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
|- -. F.
Theorem	trud 1329	Eliminate T. as an antecedent. (Contributed by Mario Carneiro, 13-Mar-2014.)
|- ( T. -> ph )   =>    |- ph
Theorem	tbtru 1330	If something is true, it outputs T.. (Contributed by Anthony Hart, 14-Aug-2011.)
|- ( ph <-> ( ph <-> T. ) )
Theorem	nbfal 1331	If something is not true, it outputs F.. (Contributed by Anthony Hart, 14-Aug-2011.)
|- ( -. ph <-> ( ph <-> F. ) )
Theorem	bitru 1332	A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
|- ph   =>    |- ( ph <-> T. )
Theorem	bifal 1333	A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
|- -. ph   =>    |- ( ph <-> F. )
Theorem	falim 1334	F. implies anything. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
|- ( F. -> ph )
Theorem	falimd 1335	F. implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ( ph /\ F. ) -> ps )
Theorem	a1tru 1336	Anything implies T.. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
|- ( ph -> T. )
Theorem	truan 1337	True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.)
|- ( ( T. /\ ph ) <-> ph )
Theorem	dfnot 1338	Given falsum, we can define the negation of a wff ph as the statement that a contradiction follows from assuming ph. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( -. ph <-> ( ph -> F. ) )
Theorem	inegd 1339	Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ( ph /\ ps ) -> F. )   =>    |- ( ph -> -. ps )
Theorem	efald 1340	Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ( ph /\ -. ps ) -> F. )   =>    |- ( ph -> ps )
Theorem	pm2.21fal 1341	If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ph -> ps )   &    |- ( ph -> -. ps )   =>    |- ( ph -> F. )
1.2.12  Truth tables Some sources define operations on true/false values using truth tables. These tables show the results of their operations for all possible combinations of true ( T.) and false ( F.). Here we show that our definitions and axioms produce equivalent results for /\ (conjunction aka logical 'and') df-an 361, \/ (disjunction aka logical inclusive 'or') df-or 360, -> (implies) wi 4, -. (not) wn 3, <-> (logical equivalence) df-bi 178, -/\ (nand aka Sheffer stroke) df-nan 1294, and \/_ (exclusive or) df-xor 1311.
Theorem	truantru 1342	A /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( T. /\ T. ) <-> T. )
Theorem	truanfal 1343	A /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( T. /\ F. ) <-> F. )
Theorem	falantru 1344	A /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( F. /\ T. ) <-> F. )
Theorem	falanfal 1345	A /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( F. /\ F. ) <-> F. )
Theorem	truortru 1346	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( T. \/ T. ) <-> T. )
Theorem	truorfal 1347	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( T. \/ F. ) <-> T. )
Theorem	falortru 1348	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( F. \/ T. ) <-> T. )
Theorem	falorfal 1349	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( F. \/ F. ) <-> F. )
Theorem	truimtru 1350	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( T. -> T. ) <-> T. )
Theorem	truimfal 1351	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( T. -> F. ) <-> F. )
Theorem	falimtru 1352	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( F. -> T. ) <-> T. )
Theorem	falimfal 1353	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( ( F. -> F. ) <-> T. )
Theorem	nottru 1354	A -. identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( -. T. <-> F. )
Theorem	notfal 1355	A -. identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( -. F. <-> T. )
Theorem	trubitru 1356	A <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( T. <-> T. ) <-> T. )
Theorem	trubifal 1357	A <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( T. <-> F. ) <-> F. )
Theorem	falbitru 1358	A <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( F. <-> T. ) <-> F. )
Theorem	falbifal 1359	A <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( F. <-> F. ) <-> T. )
Theorem	trunantru 1360	A -/\ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( T. -/\ T. ) <-> F. )
Theorem	trunanfal 1361	A -/\ identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( T. -/\ F. ) <-> T. )
Theorem	falnantru 1362	A -/\ identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( F. -/\ T. ) <-> T. )
Theorem	falnanfal 1363	A -/\ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( F. -/\ F. ) <-> T. )
Theorem	truxortru 1364	A \/_ identity. (Contributed by David A. Wheeler, 8-May-2015.)
|- ( ( T. \/_ T. ) <-> F. )
Theorem	truxorfal 1365	A \/_ identity. (Contributed by David A. Wheeler, 8-May-2015.)
|- ( ( T. \/_ F. ) <-> T. )
Theorem	falxortru 1366	A \/_ identity. (Contributed by David A. Wheeler, 9-May-2015.)
|- ( ( F. \/_ T. ) <-> T. )
Theorem	falxorfal 1367	A \/_ identity. (Contributed by David A. Wheeler, 9-May-2015.)
|- ( ( F. \/_ F. ) <-> F. )
1.2.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1
Theorem	ee22 1368	Virtual deduction rule e22 28490 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 2-May-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( ps -> th ) )   &    |- ( ch -> ( th -> ta ) )   =>    |- ( ph -> ( ps -> ta ) )
Theorem	ee12an 1369	e12an 28555 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 28-Oct-2011.) TODO: this is frequently used; come up with better label.
|- ( ph -> ps )   &    |- ( ph -> ( ch -> th ) )   &    |- ( ( ps /\ th ) -> ta )   =>    |- ( ph -> ( ch -> ta ) )
Theorem	ee23 1370	e23 28585 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( ps -> ( th -> ta ) ) )   &    |- ( ch -> ( ta -> et ) )   =>    |- ( ph -> ( ps -> ( th -> et ) ) )
Theorem	exbir 1371	Exportation implication also converting head from biconditional to conditional. This proof is exbirVD 28683 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.
|- ( ( ( ph /\ ps ) -> ( ch <-> th ) ) -> ( ph -> ( ps -> ( th -> ch ) ) ) )
Theorem	3impexp 1372	impexp 434 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ph -> ( ps -> ( ch -> th ) ) ) )
Theorem	3impexpbicom 1373	3impexp 1372 with biconditional consequent of antecedent that is commuted in consequent. Derived automatically from 3impexpVD 28686. (Contributed by Alan Sare, 31-Dec-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.
|- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )
Theorem	3impexpbicomi 1374	Deduction form of 3impexpbicom 1373. Derived automatically from 3impexpbicomiVD 28688. (Contributed by Alan Sare, 31-Dec-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.
|- ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) )   =>    |- ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) )
Theorem	ancomsimp 1375	Closed form of ancoms 440. Derived automatically from ancomsimpVD 28695. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ps /\ ph ) -> ch ) )
Theorem	exp3acom3r 1376	Export and commute antecedents. (Contributed by Alan Sare, 18-Mar-2012.)
|- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ps -> ( ch -> ( ph -> th ) ) )
Theorem	exp3acom23g 1377	Implication form of exp3acom23 1378. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.) TODO: decide if this is worth keeping.
|- ( ( ph -> ( ( ps /\ ch ) -> th ) ) <-> ( ph -> ( ch -> ( ps -> th ) ) ) )
Theorem	exp3acom23 1378	The exportation deduction exp3a 426 with commutation of the conjoined wwfs. (Contributed by Alan Sare, 22-Jul-2012.)
|- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ph -> ( ch -> ( ps -> th ) ) )
Theorem	simplbi2comg 1379	Implication form of simplbi2com 1380. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.) TODO: decide if this is worth keeping.
|- ( ( ph <-> ( ps /\ ch ) ) -> ( ch -> ( ps -> ph ) ) )
Theorem	simplbi2com 1380	A deduction eliminating a conjunct, similar to simplbi2 609. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.)
|- ( ph <-> ( ps /\ ch ) )   =>    |- ( ch -> ( ps -> ph ) )
Theorem	ee21 1381	e21 28560 without virtual deductions. (Contributed by Alan Sare, 18-Mar-2012.) (New usage is discouraged.) TODO: decide if this is worth keeping.
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> th )   &    |- ( ch -> ( th -> ta ) )   =>    |- ( ph -> ( ps -> ta ) )
Theorem	ee10 1382	e10 28513 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) TODO: this is frequently used; come up with better label.
|- ( ph -> ps )   &    |- ch   &    |- ( ps -> ( ch -> th ) )   =>    |- ( ph -> th )
Theorem	ee02 1383	e02 28516 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.) TODO: decide if this is worth keeping.
|- ph   &    |- ( ps -> ( ch -> th ) )   &    |- ( ph -> ( th -> ta ) )   =>    |- ( ps -> ( ch -> ta ) )
1.2.14  Half-adders and full adders in propositional calculus Propositional calculus deals with truth values, which can be interpreted as bits. Using this, we can define the half-adder in pure propositional calculus, and show its basic properties.
Syntax	whad 1384	Define the half adder (triple XOR). (Contributed by Mario Carneiro, 4-Sep-2016.)
wff hadd ( ph , ps , ch )
Syntax	wcad 1385	Define the half adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
wff cadd ( ph , ps , ch )
Definition	df-had 1386	Define the half adder (triple XOR). (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (hadd ( ph , ps , ch ) <-> ( ( ph \/_ ps ) \/_ ch ) )
Definition	df-cad 1387	Define the half adder carry, which is true when at least two arguments are true. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (cadd ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( ch /\ ( ph \/_ ps ) ) ) )
Theorem	hadbi123d 1388	Equality theorem for half adder. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   &    |- ( ph -> ( et <-> ze ) )   =>    |- ( ph -> (hadd ( ps , th , et ) <-> hadd ( ch , ta , ze ) ) )
Theorem	cadbi123d 1389	Equality theorem for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   &    |- ( ph -> ( et <-> ze ) )   =>    |- ( ph -> (cadd ( ps , th , et ) <-> cadd ( ch , ta , ze ) ) )
Theorem	hadbi123i 1390	Equality theorem for half adder. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   &    |- ( ta <-> et )   =>    |- (hadd ( ph , ch , ta ) <-> hadd ( ps , th , et ) )
Theorem	cadbi123i 1391	Equality theorem for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   &    |- ( ta <-> et )   =>    |- (cadd ( ph , ch , ta ) <-> cadd ( ps , th , et ) )
Theorem	hadass 1392	Associative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (hadd ( ph , ps , ch ) <-> ( ph \/_ ( ps \/_ ch ) ) )
Theorem	hadbi 1393	The half adder is the same as the triple biconditional. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (hadd ( ph , ps , ch ) <-> ( ( ph <-> ps ) <-> ch ) )
Theorem	hadcoma 1394	Commutative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (hadd ( ph , ps , ch ) <-> hadd ( ps , ph , ch ) )
Theorem	hadcomb 1395	Commutative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (hadd ( ph , ps , ch ) <-> hadd ( ph , ch , ps ) )
Theorem	hadrot 1396	Rotation law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (hadd ( ph , ps , ch ) <-> hadd ( ps , ch , ph ) )
Theorem	cador 1397	Write the adder carry in disjunctive normal form. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (cadd ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( ph /\ ch ) \/ ( ps /\ ch ) ) )
Theorem	cadan 1398	Write the adder carry in conjunctive normal form. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (cadd ( ph , ps , ch ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) /\ ( ps \/ ch ) ) )
Theorem	hadnot 1399	The half adder distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( -. hadd ( ph , ps , ch ) <-> hadd ( -. ph , -. ps , -. ch ) )
Theorem	cadnot 1400	The adder carry distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ( -. cadd ( ph , ps , ch ) <-> cadd ( -. ph , -. ps , -. ch ) )