P. 38 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3701-3800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iuneq1d 3701*	Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
|- ( ph -> A = B )   =>    |- ( ph -> U_ x e. A C = U_ x e. B C )
Theorem	iuneq12d 3702*	Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> U_ x e. A C = U_ x e. B D )
Theorem	iuneq2d 3703*	Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)
|- ( ph -> B = C )   =>    |- ( ph -> U_ x e. A B = U_ x e. A C )
Theorem	nfiunxy 3704*	Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
|- F/_ y A   &    |- F/_ y B   =>    |- F/_ y U_ x e. A B
Theorem	nfiinxy 3705*	Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
|- F/_ y A   &    |- F/_ y B   =>    |- F/_ y |^|_ x e. A B
Theorem	nfiunya 3706*	Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
|- F/_ y A   &    |- F/_ y B   =>    |- F/_ y U_ x e. A B
Theorem	nfiinya 3707*	Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
|- F/_ y A   &    |- F/_ y B   =>    |- F/_ y |^|_ x e. A B
Theorem	nfiu1 3708	Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
|- F/_ x U_ x e. A B
Theorem	nfii1 3709	Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
|- F/_ x |^|_ x e. A B
Theorem	dfiun2g 3710*	Alternate definition of indexed union when B is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( A. x e. A B e. C -> U_ x e. A B = U. { y | E. x e. A y = B } )
Theorem	dfiin2g 3711*	Alternate definition of indexed intersection when B is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)
|- ( A. x e. A B e. C -> |^|_ x e. A B = |^| { y | E. x e. A y = B } )
Theorem	dfiun2 3712*	Alternate definition of indexed union when B is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by David Abernethy, 19-Jun-2012.)
|- B e. _V   =>    |- U_ x e. A B = U. { y | E. x e. A y = B }
Theorem	dfiin2 3713*	Alternate definition of indexed intersection when B is a set. Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- B e. _V   =>    |- |^|_ x e. A B = |^| { y | E. x e. A y = B }
Theorem	dfiunv2 3714*	Define double indexed union. (Contributed by FL, 6-Nov-2013.)
|- U_ x e. A U_ y e. B C = { z | E. x e. A E. y e. B z e. C }
Theorem	cbviun 3715*	Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)
|- F/_ y B   &    |- F/_ x C   &    |- ( x = y -> B = C )   =>    |- U_ x e. A B = U_ y e. A C
Theorem	cbviin 3716*	Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- F/_ y B   &    |- F/_ x C   &    |- ( x = y -> B = C )   =>    |- |^|_ x e. A B = |^|_ y e. A C
Theorem	cbviunv 3717*	Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.)
|- ( x = y -> B = C )   =>    |- U_ x e. A B = U_ y e. A C
Theorem	cbviinv 3718*	Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.)
|- ( x = y -> B = C )   =>    |- |^|_ x e. A B = |^|_ y e. A C
Theorem	iunss 3719*	Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( U_ x e. A B C_ C <-> A. x e. A B C_ C )
Theorem	ssiun 3720*	Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( E. x e. A C C_ B -> C C_ U_ x e. A B )
Theorem	ssiun2 3721	Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( x e. A -> B C_ U_ x e. A B )
Theorem	ssiun2s 3722*	Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)
|- ( x = C -> B = D )   =>    |- ( C e. A -> D C_ U_ x e. A B )
Theorem	iunss2 3723*	A subclass condition on the members of two indexed classes C ( x ) and D ( y ) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 3632. (Contributed by NM, 9-Dec-2004.)
|- ( A. x e. A E. y e. B C C_ D -> U_ x e. A C C_ U_ y e. B D )
Theorem	iunab 3724*	The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
|- U_ x e. A { y | ph } = { y | E. x e. A ph }
Theorem	iunrab 3725*	The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
|- U_ x e. A { y e. B | ph } = { y e. B | E. x e. A ph }
Theorem	iunxdif2 3726*	Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)
|- ( x = y -> C = D )   =>    |- ( A. x e. A E. y e. ( A \ B ) C C_ D -> U_ y e. ( A \ B ) D = U_ x e. A C )
Theorem	ssiinf 3727	Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
|- F/_ x C   =>    |- ( C C_ |^|_ x e. A B <-> A. x e. A C C_ B )
Theorem	ssiin 3728*	Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)
|- ( C C_ |^|_ x e. A B <-> A. x e. A C C_ B )
Theorem	iinss 3729*	Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( E. x e. A B C_ C -> |^|_ x e. A B C_ C )
Theorem	iinss2 3730	An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)
|- ( x e. A -> |^|_ x e. A B C_ B )
Theorem	uniiun 3731*	Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)
|- U. A = U_ x e. A x
Theorem	intiin 3732*	Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
|- |^| A = |^|_ x e. A x
Theorem	iunid 3733*	An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
|- U_ x e. A { x } = A
Theorem	iun0 3734	An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- U_ x e. A (/) = (/)
Theorem	0iun 3735	An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- U_ x e. (/) A = (/)
Theorem	0iin 3736	An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
|- |^|_ x e. (/) A = _V
Theorem	viin 3737*	Indexed intersection with a universal index class. (Contributed by NM, 11-Sep-2008.)
|- |^|_ x e. _V A = { y | A. x y e. A }
Theorem	iunn0m 3738*	There is an inhabited class in an indexed collection B ( x ) iff the indexed union of them is inhabited. (Contributed by Jim Kingdon, 16-Aug-2018.)
|- ( E. x e. A E. y y e. B <-> E. y y e. U_ x e. A B )
Theorem	iinab 3739*	Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
|- |^|_ x e. A { y | ph } = { y | A. x e. A ph }
Theorem	iinrabm 3740*	Indexed intersection of a restricted class builder. (Contributed by Jim Kingdon, 16-Aug-2018.)
|- ( E. x x e. A -> |^|_ x e. A { y e. B | ph } = { y e. B | A. x e. A ph } )
Theorem	iunin2 3741*	Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3731 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
|- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C )
Theorem	iunin1 3742*	Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3731 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
|- U_ x e. A ( C i^i B ) = ( U_ x e. A C i^i B )
Theorem	iundif2ss 3743*	Indexed union of class difference. Compare to theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
|- U_ x e. A ( B \ C ) C_ ( B \ |^|_ x e. A C )
Theorem	2iunin 3744*	Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)
|- U_ x e. A U_ y e. B ( C i^i D ) = ( U_ x e. A C i^i U_ y e. B D )
Theorem	iindif2m 3745*	Indexed intersection of class difference. Compare to Theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
|- ( E. x x e. A -> |^|_ x e. A ( B \ C ) = ( B \ U_ x e. A C ) )
Theorem	iinin2m 3746*	Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
|- ( E. x x e. A -> |^|_ x e. A ( B i^i C ) = ( B i^i |^|_ x e. A C ) )
Theorem	iinin1m 3747*	Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
|- ( E. x x e. A -> |^|_ x e. A ( C i^i B ) = ( |^|_ x e. A C i^i B ) )
Theorem	elriin 3748*	Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)
|- ( B e. ( A i^i |^|_ x e. X S ) <-> ( B e. A /\ A. x e. X B e. S ) )
Theorem	riin0 3749*	Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( X = (/) -> ( A i^i |^|_ x e. X S ) = A )
Theorem	riinm 3750*	Relative intersection of an inhabited family. (Contributed by Jim Kingdon, 19-Aug-2018.)
|- ( ( A. x e. X S C_ A /\ E. x x e. X ) -> ( A i^i |^|_ x e. X S ) = |^|_ x e. X S )
Theorem	iinxsng 3751*	A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
|- ( x = A -> B = C )   =>    |- ( A e. V -> |^|_ x e. { A } B = C )
Theorem	iinxprg 3752*	Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)
|- ( x = A -> C = D )   &    |- ( x = B -> C = E )   =>    |- ( ( A e. V /\ B e. W ) -> |^|_ x e. { A , B } C = ( D i^i E ) )
Theorem	iunxsng 3753*	A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)
|- ( x = A -> B = C )   =>    |- ( A e. V -> U_ x e. { A } B = C )
Theorem	iunxsn 3754*	A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 25-Jun-2016.)
|- A e. _V   &    |- ( x = A -> B = C )   =>    |- U_ x e. { A } B = C
Theorem	iunun 3755	Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
|- U_ x e. A ( B u. C ) = ( U_ x e. A B u. U_ x e. A C )
Theorem	iunxun 3756	Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
|- U_ x e. ( A u. B ) C = ( U_ x e. A C u. U_ x e. B C )
Theorem	iunxiun 3757*	Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)
|- U_ x e. U_ y e. A B C = U_ y e. A U_ x e. B C
Theorem	iinuniss 3758*	A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33 but with equality changed to subset. (Contributed by Jim Kingdon, 19-Aug-2018.)
|- ( A u. |^| B ) C_ |^|_ x e. B ( A u. x )
Theorem	iununir 3759*	A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33 but with biconditional changed to implication. (Contributed by Jim Kingdon, 19-Aug-2018.)
|- ( ( A u. U. B ) = U_ x e. B ( A u. x ) -> ( B = (/) -> A = (/) ) )
Theorem	sspwuni 3760	Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
|- ( A C_ ~P B <-> U. A C_ B )
Theorem	pwssb 3761*	Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)
|- ( A C_ ~P B <-> A. x e. A x C_ B )
Theorem	elpwuni 3762	Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
|- ( B e. A -> ( A C_ ~P B <-> U. A = B ) )
Theorem	iinpw 3763*	The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
|- ~P |^| A = |^|_ x e. A ~P x
Theorem	iunpwss 3764*	Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.)
|- U_ x e. A ~P x C_ ~P U. A
Theorem	rintm 3765*	Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.)
|- ( ( X C_ ~P A /\ E. x x e. X ) -> ( A i^i |^| X ) = |^| X )
2.1.21  Disjointness
Syntax	wdisj 3766	Extend wff notation to include the statement that a family of classes B ( x ), for x e. A, is a disjoint family.
wff Disj x e. A B
Definition	df-disj 3767*	A collection of classes B ( x ) is disjoint when for each element y, it is in B ( x ) for at most one x. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.)
|- (Disj x e. A B <-> A. y E* x e. A y e. B )
Theorem	dfdisj2 3768*	Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
|- (Disj x e. A B <-> A. y E* x ( x e. A /\ y e. B ) )
Theorem	disjss2 3769	If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( A. x e. A B C_ C -> (Disj x e. A C -> Disj x e. A B ) )
Theorem	disjeq2 3770	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( A. x e. A B = C -> (Disj x e. A B <-> Disj x e. A C ) )
Theorem	disjeq2dv 3771*	Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( ( ph /\ x e. A ) -> B = C )   =>    |- ( ph -> (Disj x e. A B <-> Disj x e. A C ) )
Theorem	disjss1 3772*	A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( A C_ B -> (Disj x e. B C -> Disj x e. A C ) )
Theorem	disjeq1 3773*	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( A = B -> (Disj x e. A C <-> Disj x e. B C ) )
Theorem	disjeq1d 3774*	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( ph -> A = B )   =>    |- ( ph -> (Disj x e. A C <-> Disj x e. B C ) )
Theorem	disjeq12d 3775*	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> (Disj x e. A C <-> Disj x e. B D ) )
Theorem	cbvdisj 3776*	Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- F/_ y B   &    |- F/_ x C   &    |- ( x = y -> B = C )   =>    |- (Disj x e. A B <-> Disj y e. A C )
Theorem	cbvdisjv 3777*	Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
|- ( x = y -> B = C )   =>    |- (Disj x e. A B <-> Disj y e. A C )
Theorem	nfdisjv 3778*	Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.)
|- F/_ y A   &    |- F/_ y B   =>    |- F/ yDisj x e. A B
Theorem	nfdisj1 3779	Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- F/ xDisj x e. A B
Theorem	invdisj 3780*	If there is a function C ( y ) such that C ( y ) = x for all y e. B ( x ), then the sets B ( x ) for distinct x e. A are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
|- ( A. x e. A A. y e. B C = x -> Disj x e. A B )
Theorem	sndisj 3781	Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- Disj x e. A { x }
Theorem	0disj 3782	Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- Disj x e. A (/)
Theorem	disjxsn 3783*	A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- Disj x e. { A } B
Theorem	disjx0 3784	An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- Disj x e. (/) B
2.1.22  Binary relations
Syntax	wbr 3785	Extend wff notation to include the general binary relation predicate. Note that the syntax is simply three class symbols in a row. Since binary relations are the only possible wff expressions consisting of three class expressions in a row, the syntax is unambiguous.
wff A R B
Definition	df-br 3786	Define a general binary relation. Note that the syntax is simply three class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29 generalized to arbitrary classes. This definition of relations is well-defined, although not very meaningful, when classes A and/or B are proper classes (i.e. are not sets). On the other hand, we often find uses for this definition when R is a proper class (see for example iprc 4618). (Contributed by NM, 31-Dec-1993.)
|- ( A R B <-> <. A , B >. e. R )
Theorem	breq 3787	Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
|- ( R = S -> ( A R B <-> A S B ) )
Theorem	breq1 3788	Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
|- ( A = B -> ( A R C <-> B R C ) )
Theorem	breq2 3789	Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
|- ( A = B -> ( C R A <-> C R B ) )
Theorem	breq12 3790	Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )
Theorem	breqi 3791	Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
|- R = S   =>    |- ( A R B <-> A S B )
Theorem	breq1i 3792	Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- A = B   =>    |- ( A R C <-> B R C )
Theorem	breq2i 3793	Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- A = B   =>    |- ( C R A <-> C R B )
Theorem	breq12i 3794	Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
|- A = B   &    |- C = D   =>    |- ( A R C <-> B R D )
Theorem	breq1d 3795	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A R C <-> B R C ) )
Theorem	breqd 3796	Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C A D <-> C B D ) )
Theorem	breq2d 3797	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C R A <-> C R B ) )
Theorem	breq12d 3798	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A R C <-> B R D ) )
Theorem	breq123d 3799	Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
|- ( ph -> A = B )   &    |- ( ph -> R = S )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A R C <-> B S D ) )
Theorem	breqan12d 3800	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ph /\ ps ) -> ( A R C <-> B R D ) )