P. 36 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3501-3600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	disjel 3501	A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)
|- ( ( ( A i^i B ) = (/) /\ C e. A ) -> -. C e. B )
Theorem	disj2 3502	Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
|- ( ( A i^i B ) = (/) <-> A C_ ( _V \ B ) )
Theorem	ssdisj 3503	Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)
|- ( ( A C_ B /\ ( B i^i C ) = (/) ) -> ( A i^i C ) = (/) )
Theorem	undisj1 3504	The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)
|- ( ( ( A i^i C ) = (/) /\ ( B i^i C ) = (/) ) <-> ( ( A u. B ) i^i C ) = (/) )
Theorem	undisj2 3505	The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)
|- ( ( ( A i^i B ) = (/) /\ ( A i^i C ) = (/) ) <-> ( A i^i ( B u. C ) ) = (/) )
Theorem	ssindif0im 3506	Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
|- ( A C_ B -> ( A i^i ( _V \ B ) ) = (/) )
Theorem	inelcm 3507	The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
|- ( ( A e. B /\ A e. C ) -> ( B i^i C ) =/= (/) )
Theorem	minel 3508	A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)
|- ( ( A e. B /\ ( C i^i B ) = (/) ) -> -. A e. C )
Theorem	undif4 3509	Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ( A i^i C ) = (/) -> ( A u. ( B \ C ) ) = ( ( A u. B ) \ C ) )
Theorem	disjssun 3510	Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ( A i^i B ) = (/) -> ( A C_ ( B u. C ) <-> A C_ C ) )
Theorem	ssdif0im 3511	Subclass implies empty difference. One direction of Exercise 7 of [TakeutiZaring] p. 22. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 2-Aug-2018.)
|- ( A C_ B -> ( A \ B ) = (/) )
Theorem	vdif0im 3512	Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)
|- ( A = _V -> ( _V \ A ) = (/) )
Theorem	difrab0eqim 3513*	If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Jim Kingdon, 3-Aug-2018.)
|- ( V = { x e. V | ph } -> ( V \ { x e. V | ph } ) = (/) )
Theorem	inssdif0im 3514	Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)
|- ( ( A i^i B ) C_ C -> ( A i^i ( B \ C ) ) = (/) )
Theorem	difid 3515	The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. (Contributed by NM, 22-Apr-2004.)
|- ( A \ A ) = (/)
Theorem	difidALT 3516	The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. Alternate proof of difid 3515. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( A \ A ) = (/)
Theorem	dif0 3517	The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
|- ( A \ (/) ) = A
Theorem	0dif 3518	The difference between the empty set and a class. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
|- ( (/) \ A ) = (/)
Theorem	disjdif 3519	A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)
|- ( A i^i ( B \ A ) ) = (/)
Theorem	difin0 3520	The difference of a class from its intersection is empty. Theorem 37 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ( A i^i B ) \ B ) = (/)
Theorem	undif1ss 3521	Absorption of difference by union. In classical logic, as Theorem 35 of [Suppes] p. 29, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
|- ( ( A \ B ) u. B ) C_ ( A u. B )
Theorem	undif2ss 3522	Absorption of difference by union. In classical logic, as in Part of proof of Corollary 6K of [Enderton] p. 144, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
|- ( A u. ( B \ A ) ) C_ ( A u. B )
Theorem	undifabs 3523	Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
|- ( A u. ( A \ B ) ) = A
Theorem	inundifss 3524	The intersection and class difference of a class with another class are contained in the original class. In classical logic we'd be able to make a stronger statement: that everything in the original class is in the intersection or the difference (that is, this theorem would be equality rather than subset). (Contributed by Jim Kingdon, 4-Aug-2018.)
|- ( ( A i^i B ) u. ( A \ B ) ) C_ A
Theorem	disjdif2 3525	The difference of a class and a class disjoint from it is the original class. (Contributed by BJ, 21-Apr-2019.)
|- ( ( A i^i B ) = (/) -> ( A \ B ) = A )
Theorem	difun2 3526	Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
|- ( ( A u. B ) \ B ) = ( A \ B )
Theorem	undifss 3527	Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)
|- ( A C_ B <-> ( A u. ( B \ A ) ) C_ B )
Theorem	ssdifin0 3528	A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
|- ( A C_ ( B \ C ) -> ( A i^i C ) = (/) )
Theorem	ssdifeq0 3529	A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)
|- ( A C_ ( B \ A ) <-> A = (/) )
Theorem	ssundifim 3530	A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
|- ( A C_ ( B u. C ) -> ( A \ B ) C_ C )
Theorem	difdifdirss 3531	Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
|- ( ( A \ B ) \ C ) C_ ( ( A \ C ) \ ( B \ C ) )
Theorem	uneqdifeqim 3532	Two ways that A and B can "partition" C (when A and B don't overlap and A is a part of C). In classical logic, the second implication would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
|- ( ( A C_ C /\ ( A i^i B ) = (/) ) -> ( ( A u. B ) = C -> ( C \ A ) = B ) )
Theorem	r19.2m 3533*	Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1649). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) (Revised by Jim Kingdon, 7-Apr-2023.)
|- ( ( E. y y e. A /\ A. x e. A ph ) -> E. x e. A ph )
Theorem	r19.2mOLD 3534*	Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1649). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) Obsolete version of r19.2m 3533 as of 7-Apr-2023. (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( E. x x e. A /\ A. x e. A ph ) -> E. x e. A ph )
Theorem	r19.3rm 3535*	Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)
|- F/ x ph   =>    |- ( E. y y e. A -> ( ph <-> A. x e. A ph ) )
Theorem	r19.28m 3536*	Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
|- F/ x ph   =>    |- ( E. x x e. A -> ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) )
Theorem	r19.3rmv 3537*	Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.)
|- ( E. y y e. A -> ( ph <-> A. x e. A ph ) )
Theorem	r19.9rmv 3538*	Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)
|- ( E. y y e. A -> ( ph <-> E. x e. A ph ) )
Theorem	r19.28mv 3539*	Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
|- ( E. x x e. A -> ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) )
Theorem	r19.45mv 3540*	Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
|- ( E. x x e. A -> ( E. x e. A ( ph \/ ps ) <-> ( ph \/ E. x e. A ps ) ) )
Theorem	r19.44mv 3541*	Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
|- ( E. y y e. A -> ( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ ps ) ) )
Theorem	r19.27m 3542*	Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
|- F/ x ps   =>    |- ( E. x x e. A -> ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ ps ) ) )
Theorem	r19.27mv 3543*	Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
|- ( E. x x e. A -> ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ ps ) ) )
Theorem	rzal 3544*	Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( A = (/) -> A. x e. A ph )
Theorem	rexn0 3545*	Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3546). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
|- ( E. x e. A ph -> A =/= (/) )
Theorem	rexm 3546*	Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)
|- ( E. x e. A ph -> E. x x e. A )
Theorem	ralidm 3547*	Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
|- ( A. x e. A A. x e. A ph <-> A. x e. A ph )
Theorem	ral0 3548	Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
|- A. x e. (/) ph
Theorem	ralf0 3549*	The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
|- -. ph   =>    |- ( A. x e. A ph <-> A = (/) )
Theorem	ralm 3550	Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)
|- ( ( E. x x e. A -> A. x e. A ph ) <-> A. x e. A ph )
Theorem	raaanlem 3551*	Special case of raaan 3552 where A is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
|- F/ y ph   &    |- F/ x ps   =>    |- ( E. x x e. A -> ( A. x e. A A. y e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. y e. A ps ) ) )
Theorem	raaan 3552*	Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)
|- F/ y ph   &    |- F/ x ps   =>    |- ( A. x e. A A. y e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. y e. A ps ) )
Theorem	raaanv 3553*	Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
|- ( A. x e. A A. y e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. y e. A ps ) )
Theorem	sbss 3554*	Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
|- ( [ y / x ] x C_ A <-> y C_ A )
Theorem	sbcssg 3555	Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)
|- ( A e. V -> ( [. A / x ]. B C_ C <-> [_ A / x ]_ B C_ [_ A / x ]_ C ) )
Theorem	dcun 3556	The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.)
|- ( ph -> DECID k e. A )   &    |- ( ph -> DECID k e. B )   =>    |- ( ph -> DECID k e. ( A u. B ) )
2.1.15  Conditional operator
Syntax	cif 3557	Extend class notation to include the conditional operator. See df-if 3558 for a description. (In older databases this was denoted "ded".)
class if ( ph , A , B )
Definition	df-if 3558*	Define the conditional operator. Read if ( ph , A , B ) as "if ph then A else B". See iftrue 3562 and iffalse 3565 for its values. In mathematical literature, this operator is rarely defined formally but is implicit in informal definitions such as "let f(x)=0 if x=0 and 1/x otherwise." In the absence of excluded middle, this will tend to be useful where ph is decidable (in the sense of df-dc 836). (Contributed by NM, 15-May-1999.)
|- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
Theorem	dfif6 3559*	An alternate definition of the conditional operator df-if 3558 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
|- if ( ph , A , B ) = ( { x e. A | ph } u. { x e. B | -. ph } )
Theorem	ifeq1 3560	Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ( A = B -> if ( ph , A , C ) = if ( ph , B , C ) )
Theorem	ifeq2 3561	Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ( A = B -> if ( ph , C , A ) = if ( ph , C , B ) )
Theorem	iftrue 3562	Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ph -> if ( ph , A , B ) = A )
Theorem	iftruei 3563	Inference associated with iftrue 3562. (Contributed by BJ, 7-Oct-2018.)
|- ph   =>    |- if ( ph , A , B ) = A
Theorem	iftrued 3564	Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> ch )   =>    |- ( ph -> if ( ch , A , B ) = A )
Theorem	iffalse 3565	Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
|- ( -. ph -> if ( ph , A , B ) = B )
Theorem	iffalsei 3566	Inference associated with iffalse 3565. (Contributed by BJ, 7-Oct-2018.)
|- -. ph   =>    |- if ( ph , A , B ) = B
Theorem	iffalsed 3567	Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> -. ch )   =>    |- ( ph -> if ( ch , A , B ) = B )
Theorem	ifnefalse 3568	When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs versus applying iffalse 3565 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
|- ( A =/= B -> if ( A = B , C , D ) = D )
Theorem	ifsbdc 3569	Distribute a function over an if-clause. (Contributed by Jim Kingdon, 1-Jan-2022.)
|- ( if ( ph , A , B ) = A -> C = D )   &    |- ( if ( ph , A , B ) = B -> C = E )   =>    |- (DECID ph -> C = if ( ph , D , E ) )
Theorem	dfif3 3570*	Alternate definition of the conditional operator df-if 3558. Note that ph is independent of x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- C = { x | ph }   =>    |- if ( ph , A , B ) = ( ( A i^i C ) u. ( B i^i ( _V \ C ) ) )
Theorem	ifssun 3571	A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.)
|- if ( ph , A , B ) C_ ( A u. B )
Theorem	ifidss 3572	A conditional class whose two alternatives are equal is included in that alternative. With excluded middle, we can prove it is equal to it. (Contributed by BJ, 15-Aug-2024.)
|- if ( ph , A , A ) C_ A
Theorem	ifeq12 3573	Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
|- ( ( A = B /\ C = D ) -> if ( ph , A , C ) = if ( ph , B , D ) )
Theorem	ifeq1d 3574	Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
|- ( ph -> A = B )   =>    |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )
Theorem	ifeq2d 3575	Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
|- ( ph -> A = B )   =>    |- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) )
Theorem	ifeq12d 3576	Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> if ( ps , A , C ) = if ( ps , B , D ) )
Theorem	ifbi 3577	Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
|- ( ( ph <-> ps ) -> if ( ph , A , B ) = if ( ps , A , B ) )
Theorem	ifbid 3578	Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> if ( ps , A , B ) = if ( ch , A , B ) )
Theorem	ifbieq1d 3579	Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> A = B )   =>    |- ( ph -> if ( ps , A , C ) = if ( ch , B , C ) )
Theorem	ifbieq2i 3580	Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph <-> ps )   &    |- A = B   =>    |- if ( ph , C , A ) = if ( ps , C , B )
Theorem	ifbieq2d 3581	Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> A = B )   =>    |- ( ph -> if ( ps , C , A ) = if ( ch , C , B ) )
Theorem	ifbieq12i 3582	Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)
|- ( ph <-> ps )   &    |- A = C   &    |- B = D   =>    |- if ( ph , A , B ) = if ( ps , C , D )
Theorem	ifbieq12d 3583	Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> if ( ps , A , B ) = if ( ch , C , D ) )
Theorem	nfifd 3584	Deduction version of nfif 3585. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- ( ph -> F/ x ps )   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> F/_ x if ( ps , A , B ) )
Theorem	nfif 3585	Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- F/ x ph   &    |- F/_ x A   &    |- F/_ x B   =>    |- F/_ x if ( ph , A , B )
Theorem	ifcldadc 3586	Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.)
|- ( ( ph /\ ps ) -> A e. C )   &    |- ( ( ph /\ -. ps ) -> B e. C )   &    |- ( ph -> DECID ps )   =>    |- ( ph -> if ( ps , A , B ) e. C )
Theorem	ifeq1dadc 3587	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( ph /\ ps ) -> A = B )   &    |- ( ph -> DECID ps )   =>    |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )
Theorem	ifeq2dadc 3588	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( ph /\ -. ps ) -> A = B )   &    |- ( ph -> DECID ps )   =>    |- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) )
Theorem	ifbothdadc 3589	A formula th containing a decidable conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 3-Jun-2022.)
|- ( A = if ( ph , A , B ) -> ( ps <-> th ) )   &    |- ( B = if ( ph , A , B ) -> ( ch <-> th ) )   &    |- ( ( et /\ ph ) -> ps )   &    |- ( ( et /\ -. ph ) -> ch )   &    |- ( et -> DECID ph )   =>    |- ( et -> th )
Theorem	ifbothdc 3590	A wff th containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.)
|- ( A = if ( ph , A , B ) -> ( ps <-> th ) )   &    |- ( B = if ( ph , A , B ) -> ( ch <-> th ) )   =>    |- ( ( ps /\ ch /\ DECID ph ) -> th )
Theorem	ifiddc 3591	Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
|- (DECID ph -> if ( ph , A , A ) = A )
Theorem	eqifdc 3592	Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.)
|- (DECID ph -> ( A = if ( ph , B , C ) <-> ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) ) )
Theorem	ifcldcd 3593	Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.)
|- ( ph -> A e. C )   &    |- ( ph -> B e. C )   &    |- ( ph -> DECID ps )   =>    |- ( ph -> if ( ps , A , B ) e. C )
Theorem	ifnotdc 3594	Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)
|- (DECID ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )
Theorem	ifandc 3595	Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- (DECID ph -> if ( ( ph /\ ps ) , A , B ) = if ( ph , if ( ps , A , B ) , B ) )
Theorem	ifordc 3596	Rewrite a disjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- (DECID ph -> if ( ( ph \/ ps ) , A , B ) = if ( ph , A , if ( ps , A , B ) ) )
Theorem	ifmdc 3597	If a conditional class is inhabited, then the condition is decidable. This shows that conditionals are not very useful unless one can prove the condition decidable. (Contributed by BJ, 24-Sep-2022.)
|- ( A e. if ( ph , B , C ) -> DECID ph )
Theorem	ifnetruedc 3598	Deduce truth from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
|- ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) -> ph )
Theorem	ifnefals 3599	Deduce falsehood from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
|- ( ( A =/= B /\ if ( ph , A , B ) = B ) -> -. ph )
Theorem	ifnebibdc 3600	The converse of ifbi 3577 holds if the two values are not equal. (Contributed by Thierry Arnoux, 20-Feb-2025.)
|- ( (DECID ph /\ DECID ps /\ A =/= B ) -> ( if ( ph , A , B ) = if ( ps , A , B ) <-> ( ph <-> ps ) ) )