P. 34 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3301-3400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	neq0r 3301*	An inhabited class is nonempty. See n0rf 3299 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
|- ( E. x x e. A -> -. A = (/) )
Theorem	reximdva0m 3302*	Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)
|- ( ( ph /\ x e. A ) -> ps )   =>    |- ( ( ph /\ E. x x e. A ) -> E. x e. A ps )
Theorem	n0mmoeu 3303*	A case of equivalence of "at most one" and "only one". If a class is inhabited, that class having at most one element is equivalent to it having only one element. (Contributed by Jim Kingdon, 31-Jul-2018.)
|- ( E. x x e. A -> ( E* x x e. A <-> E! x x e. A ) )
Theorem	rex0 3304	Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
|- -. E. x e. (/) ph
Theorem	eq0 3305*	The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)
|- ( A = (/) <-> A. x -. x e. A )
Theorem	eqv 3306*	The universe contains every set. (Contributed by NM, 11-Sep-2006.)
|- ( A = _V <-> A. x x e. A )
Theorem	notm0 3307*	A class is not inhabited if and only if it is empty. (Contributed by Jim Kingdon, 1-Jul-2022.)
|- ( -. E. x x e. A <-> A = (/) )
Theorem	nel0 3308*	From the general negation of membership in A, infer that A is the empty set. (Contributed by BJ, 6-Oct-2018.)
|- -. x e. A   =>    |- A = (/)
Theorem	0el 3309*	Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
|- ( (/) e. A <-> E. x e. A A. y -. y e. x )
Theorem	abvor0dc 3310*	The class builder of a decidable proposition not containing the abstraction variable is either the universal class or the empty set. (Contributed by Jim Kingdon, 1-Aug-2018.)
|- (DECID ph -> ( { x | ph } = _V \/ { x | ph } = (/) ) )
Theorem	abn0r 3311	Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
|- ( E. x ph -> { x | ph } =/= (/) )
Theorem	abn0m 3312*	Inhabited class abstraction. (Contributed by Jim Kingdon, 8-Jul-2022.)
|- ( E. y y e. { x | ph } <-> E. x ph )
Theorem	rabn0r 3313	Nonempty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
|- ( E. x e. A ph -> { x e. A | ph } =/= (/) )
Theorem	rabn0m 3314*	Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)
|- ( E. y y e. { x e. A | ph } <-> E. x e. A ph )
Theorem	rab0 3315	Any restricted class abstraction restricted to the empty set is empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- { x e. (/) | ph } = (/)
Theorem	rabeq0 3316	Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)
|- ( { x e. A | ph } = (/) <-> A. x e. A -. ph )
Theorem	abeq0 3317	Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)
|- ( { x | ph } = (/) <-> A. x -. ph )
Theorem	rabxmdc 3318*	Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)
|- ( A. xDECID ph -> A = ( { x e. A | ph } u. { x e. A | -. ph } ) )
Theorem	rabnc 3319*	Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
|- ( { x e. A | ph } i^i { x e. A | -. ph } ) = (/)
Theorem	un0 3320	The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
|- ( A u. (/) ) = A
Theorem	in0 3321	The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
|- ( A i^i (/) ) = (/)
Theorem	0in 3322	The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( (/) i^i A ) = (/)
Theorem	inv1 3323	The intersection of a class with the universal class is itself. Exercise 4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
|- ( A i^i _V ) = A
Theorem	unv 3324	The union of a class with the universal class is the universal class. Exercise 4.10(l) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
|- ( A u. _V ) = _V
Theorem	0ss 3325	The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
|- (/) C_ A
Theorem	ss0b 3326	Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)
|- ( A C_ (/) <-> A = (/) )
Theorem	ss0 3327	Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
|- ( A C_ (/) -> A = (/) )
Theorem	sseq0 3328	A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ( A C_ B /\ B = (/) ) -> A = (/) )
Theorem	ssn0 3329	A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
|- ( ( A C_ B /\ A =/= (/) ) -> B =/= (/) )
Theorem	abf 3330	A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
|- -. ph   =>    |- { x | ph } = (/)
Theorem	eq0rdv 3331*	Deduction for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
|- ( ph -> -. x e. A )   =>    |- ( ph -> A = (/) )
Theorem	csbprc 3332	The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.)
|- ( -. A e. _V -> [_ A / x ]_ B = (/) )
Theorem	un00 3333	Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
|- ( ( A = (/) /\ B = (/) ) <-> ( A u. B ) = (/) )
Theorem	vss 3334	Only the universal class has the universal class as a subclass. (Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( _V C_ A <-> A = _V )
Theorem	disj 3335*	Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
|- ( ( A i^i B ) = (/) <-> A. x e. A -. x e. B )
Theorem	disjr 3336*	Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
|- ( ( A i^i B ) = (/) <-> A. x e. B -. x e. A )
Theorem	disj1 3337*	Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)
|- ( ( A i^i B ) = (/) <-> A. x ( x e. A -> -. x e. B ) )
Theorem	reldisj 3338	Two ways of saying that two classes are disjoint, using the complement of B relative to a universe C. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( A C_ C -> ( ( A i^i B ) = (/) <-> A C_ ( C \ B ) ) )
Theorem	disj3 3339	Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
|- ( ( A i^i B ) = (/) <-> A = ( A \ B ) )
Theorem	disjne 3340	Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ( ( A i^i B ) = (/) /\ C e. A /\ D e. B ) -> C =/= D )
Theorem	disjel 3341	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 3342	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 3343	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 3344	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 3345	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 3346	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 3347	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 3348	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 3349	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 3350	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 3351	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 3352	Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)
|- ( A = _V -> ( _V \ A ) = (/) )
Theorem	difrab0eqim 3353*	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 3354	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 3355	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 3356	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 3355. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( A \ A ) = (/)
Theorem	dif0 3357	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 3358	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 3359	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 3360	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 3361	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 3362	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 3363	Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
|- ( A u. ( A \ B ) ) = A
Theorem	inundifss 3364	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 3365	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 3366	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 3367	Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)
|- ( A C_ B <-> ( A u. ( B \ A ) ) C_ B )
Theorem	ssdifin0 3368	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 3369	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 3370	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 3371	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 3372	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 3373*	Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1575). The restricted version 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 ) -> E. x e. A ph )
Theorem	r19.3rm 3374*	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 3375*	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 3376*	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 3377*	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 3378*	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 3379*	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 3380*	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 3381*	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 3382*	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 3383*	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 3384*	Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3385). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
|- ( E. x e. A ph -> A =/= (/) )
Theorem	rexm 3385*	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 3386*	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 3387	Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
|- A. x e. (/) ph
Theorem	rgenm 3388*	Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)
|- ( ( E. x x e. A /\ x e. A ) -> ph )   =>    |- A. x e. A ph
Theorem	ralf0 3389*	The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
|- -. ph   =>    |- ( A. x e. A ph <-> A = (/) )
Theorem	ralm 3390	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 3391*	Special case of raaan 3392 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 3392*	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 3393*	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 3394*	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 3395	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 3396	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 3397	Extend class notation to include the conditional operator. See df-if 3398 for a description. (In older databases this was denoted "ded".)
class if ( ph , A , B )
Definition	df-if 3398*	Define the conditional operator. Read if ( ph , A , B ) as "if ph then A else B." See iftrue 3402 and iffalse 3405 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 782). (Contributed by NM, 15-May-1999.)
|- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
Theorem	dfif6 3399*	An alternate definition of the conditional operator df-if 3398 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 3400	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 ) )