P. 14 of Theorem List - New Foundations Explorer

Theorem List for New Foundations Explorer - 1301-1400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nanbi1d 1301	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 1302	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 1303	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 1304	Extend wff definition to include exclusive disjunction ('xor').
wff (ph \/_ ps)
Definition	df-xor 1305	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 1319) and the constant false F. (df-fal 1320), we will be able to prove these truth table values: (( T. \/_ T. ) <-> F. ) (truxortru 1358), (( T. \/_ F. ) <-> T. ) (truxorfal 1359), (( F. \/_ T. ) <-> T. ) (falxortru 1360), and (( F. \/_ F. ) <-> F. ) (falxorfal 1361). Contrast with /\ (df-an 360), \/ (df-or 359), -> (wi 4), and -/\ (df-nan 1288) . (Contributed by FL, 22-Nov-2010.)
|- ((ph \/_ ps) <-> -. (ph <-> ps))
Theorem	xnor 1306	Two ways to write XNOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ((ph <-> ps) <-> -. (ph \/_ ps))
Theorem	xorcom 1307	\/_ is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ((ph \/_ ps) <-> (ps \/_ ph))
Theorem	xorass 1308	\/_ is associative. (Contributed by FL, 22-Nov-2010.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
|- (((ph \/_ ps) \/_ ch) <-> (ph \/_ (ps \/_ ch)))
Theorem	excxor 1309	This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.)
|- ((ph \/_ ps) <-> ((ph /\ -. ps) \/ (-. ph /\ ps)))
Theorem	xor2 1310	Two ways to express "exclusive or." (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ((ph \/_ ps) <-> ((ph \/ ps) /\ -. (ph /\ ps)))
Theorem	xorneg1 1311	\/_ is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ((-. ph \/_ ps) <-> -. (ph \/_ ps))
Theorem	xorneg2 1312	\/_ is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ((ph \/_ -. ps) <-> -. (ph \/_ ps))
Theorem	xorneg 1313	\/_ is unchanged under negation of both arguments. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ((-. ph \/_ -. ps) <-> (ph \/_ ps))
Theorem	xorbi12i 1314	Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (ph <-> ps)   &   |- (ch <-> th)   =>   |- ((ph \/_ ch) <-> (ps \/_ th))
Theorem	xorbi12d 1315	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 1316	T. is a wff.
wff T.
Syntax	wfal 1317	F. is a wff.
wff F.
Theorem	trujust 1318	Soundness justification theorem for df-tru 1319. (Contributed by Mario Carneiro, 17-Nov-2013.)
|- ((ph <-> ph) <-> (ps <-> ps))
Definition	df-tru 1319	Definition of T., a tautology. T. is a constant true. In this definition biid 227 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 1320	Definition of F., a contradiction. F. is a constant false. (Contributed by Anthony Hart, 22-Oct-2010.)
|- ( F. <-> -. T. )
Theorem	tru 1321	T. is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
|- T.
Theorem	fal 1322	F. is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
|- -. F.
Theorem	trud 1323	Eliminate T. as an antecedent. (Contributed by Mario Carneiro, 13-Mar-2014.)
|- ( T. -> ph)   =>   |- ph
Theorem	tbtru 1324	If something is true, it outputs T.. (Contributed by Anthony Hart, 14-Aug-2011.)
|- (ph <-> (ph <-> T. ))
Theorem	nbfal 1325	If something is not true, it outputs F.. (Contributed by Anthony Hart, 14-Aug-2011.)
|- (-. ph <-> (ph <-> F. ))
Theorem	bitru 1326	A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
|- ph   =>   |- (ph <-> T. )
Theorem	bifal 1327	A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
|- -. ph   =>   |- (ph <-> F. )
Theorem	falim 1328	F. implies anything. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
|- ( F. -> ph)
Theorem	falimd 1329	F. implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ((ph /\ F. ) -> ps)
Theorem	a1tru 1330	Anything implies T.. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
|- (ph -> T. )
Theorem	truan 1331	True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.)
|- (( T. /\ ph) <-> ph)
Theorem	dfnot 1332	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 1333	Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ((ph /\ ps) -> F. )   =>   |- (ph -> -. ps)
Theorem	efald 1334	Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ((ph /\ -. ps) -> F. )   =>   |- (ph -> ps)
Theorem	pm2.21fal 1335	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 360, \/ (disjunction aka logical inclusive 'or') df-or 359, -> (implies) wi 4, -. (not) wn 3, <-> (logical equivalence) df-bi 177, -/\ (nand aka Sheffer stroke) df-nan 1288, and \/_ (exclusive or) df-xor 1305.
Theorem	truantru 1336	A /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (( T. /\ T. ) <-> T. )
Theorem	truanfal 1337	A /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (( T. /\ F. ) <-> F. )
Theorem	falantru 1338	A /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (( F. /\ T. ) <-> F. )
Theorem	falanfal 1339	A /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (( F. /\ F. ) <-> F. )
Theorem	truortru 1340	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- (( T. \/ T. ) <-> T. )
Theorem	truorfal 1341	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (( T. \/ F. ) <-> T. )
Theorem	falortru 1342	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (( F. \/ T. ) <-> T. )
Theorem	falorfal 1343	A \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- (( F. \/ F. ) <-> F. )
Theorem	truimtru 1344	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (( T. -> T. ) <-> T. )
Theorem	truimfal 1345	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- (( T. -> F. ) <-> F. )
Theorem	falimtru 1346	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (( F. -> T. ) <-> T. )
Theorem	falimfal 1347	A -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (( F. -> F. ) <-> T. )
Theorem	nottru 1348	A -. identity. (Contributed by Anthony Hart, 22-Oct-2010.)
|- (-. T. <-> F. )
Theorem	notfal 1349	A -. identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- (-. F. <-> T. )
Theorem	trubitru 1350	A <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- (( T. <-> T. ) <-> T. )
Theorem	trubifal 1351	A <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- (( T. <-> F. ) <-> F. )
Theorem	falbitru 1352	A <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- (( F. <-> T. ) <-> F. )
Theorem	falbifal 1353	A <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- (( F. <-> F. ) <-> T. )
Theorem	trunantru 1354	A -/\ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- (( T. -/\ T. ) <-> F. )
Theorem	trunanfal 1355	A -/\ identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- (( T. -/\ F. ) <-> T. )
Theorem	falnantru 1356	A -/\ identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- (( F. -/\ T. ) <-> T. )
Theorem	falnanfal 1357	A -/\ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- (( F. -/\ F. ) <-> T. )
Theorem	truxortru 1358	A \/_ identity. (Contributed by David A. Wheeler, 8-May-2015.)
|- (( T. \/_ T. ) <-> F. )
Theorem	truxorfal 1359	A \/_ identity. (Contributed by David A. Wheeler, 8-May-2015.)
|- (( T. \/_ F. ) <-> T. )
Theorem	falxortru 1360	A \/_ identity. (Contributed by David A. Wheeler, 9-May-2015.)
|- (( F. \/_ T. ) <-> T. )
Theorem	falxorfal 1361	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 1362	Virtual deduction rule e22 in set.mm 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 1363	e12an in set.mm 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 1364	e23 in set.mm 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 1365	Exportation implication also converting head from biconditional to conditional. This proof is exbirVD in set.mm 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 1366	impexp 433 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.)
|- (((ph /\ ps /\ ch) -> th) <-> (ph -> (ps -> (ch -> th))))
Theorem	3impexpbicom 1367	3impexp 1366 with biconditional consequent of antecedent that is commuted in consequent. Derived automatically from 3impexpVD in set.mm. (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 1368	Deduction form of 3impexpbicom 1367. Derived automatically from 3impexpbicomiVD in set.mm. (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 1369	Closed form of ancoms 439. Derived automatically from ancomsimpVD in set.mm. (Contributed by Alan Sare, 31-Dec-2011.)
|- (((ph /\ ps) -> ch) <-> ((ps /\ ph) -> ch))
Theorem	exp3acom3r 1370	Export and commute antecedents. (Contributed by Alan Sare, 18-Mar-2012.)
|- (ph -> ((ps /\ ch) -> th))   =>   |- (ps -> (ch -> (ph -> th)))
Theorem	exp3acom23g 1371	Implication form of exp3acom23 1372. (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 1372	The exportation deduction exp3a 425 with commutation of the conjoined wwfs. (Contributed by Alan Sare, 22-Jul-2012.)
|- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> (ch -> (ps -> th)))
Theorem	simplbi2comg 1373	Implication form of simplbi2com 1374. (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 1374	A deduction eliminating a conjunct, similar to simplbi2 608. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.)
|- (ph <-> (ps /\ ch))   =>   |- (ch -> (ps -> ph))
Theorem	ee21 1375	e21 in set.mm 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 1376	e10 in set.mm 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 1377	e02 in set.mm 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 1378	Define the half adder (triple XOR). (Contributed by Mario Carneiro, 4-Sep-2016.)
wff hadd(ph, ps, ch)
Syntax	wcad 1379	Define the half adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
wff cadd(ph, ps, ch)
Definition	df-had 1380	Define the half adder (triple XOR). (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (hadd(ph, ps, ch) <-> ((ph \/_ ps) \/_ ch))
Definition	df-cad 1381	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 1382	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 1383	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 1384	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 1385	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 1386	Associative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (hadd(ph, ps, ch) <-> (ph \/_ (ps \/_ ch)))
Theorem	hadbi 1387	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 1388	Commutative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (hadd(ph, ps, ch) <-> hadd(ps, ph, ch))
Theorem	hadcomb 1389	Commutative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (hadd(ph, ps, ch) <-> hadd(ph, ch, ps))
Theorem	hadrot 1390	Rotation law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (hadd(ph, ps, ch) <-> hadd(ps, ch, ph))
Theorem	cador 1391	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 1392	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 1393	The half adder distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (-. hadd(ph, ps, ch) <-> hadd(-. ph, -. ps, -. ch))
Theorem	cadnot 1394	The adder carry distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (-. cadd(ph, ps, ch) <-> cadd(-. ph, -. ps, -. ch))
Theorem	cadcoma 1395	Commutative law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (cadd(ph, ps, ch) <-> cadd(ps, ph, ch))
Theorem	cadcomb 1396	Commutative law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (cadd(ph, ps, ch) <-> cadd(ph, ch, ps))
Theorem	cadrot 1397	Rotation law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- (cadd(ph, ps, ch) <-> cadd(ps, ch, ph))
Theorem	cad1 1398	If one parameter is true, the adder carry is true exactly when at least one of the other parameters is true. (Contributed by Mario Carneiro, 8-Sep-2016.)
|- (ch -> (cadd(ph, ps, ch) <-> (ph \/ ps)))
Theorem	cad11 1399	If two parameters are true, the adder carry is true. (Contributed by Mario Carneiro, 4-Sep-2016.)
|- ((ph /\ ps) -> cadd(ph, ps, ch))
Theorem	cad0 1400	If one parameter is false, the adder carry is true exactly when both of the other two parameters are true. (Contributed by Mario Carneiro, 8-Sep-2016.)
|- (-. ch -> (cadd(ph, ps, ch) <-> (ph /\ ps)))