P. 39 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3801-3900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	prprc2 3801	A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)
|- ( -. B e. _V -> { A , B } = { A } )
Theorem	prprc 3802	An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)
|- ( ( -. A e. _V /\ -. B e. _V ) -> { A , B } = (/) )
Theorem	tpid1 3803	One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- A e. _V   =>    |- A e. { A , B , C }
Theorem	tpid1g 3804	Closed theorem form of tpid1 3803. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
|- ( A e. B -> A e. { A , C , D } )
Theorem	tpid2 3805	One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- B e. _V   =>    |- B e. { A , B , C }
Theorem	tpid2g 3806	Closed theorem form of tpid2 3805. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
|- ( A e. B -> A e. { C , A , D } )
Theorem	tpid3g 3807	Closed theorem form of tpid3 3808. (Contributed by Alan Sare, 24-Oct-2011.)
|- ( A e. B -> A e. { C , D , A } )
Theorem	tpid3 3808	One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- C e. _V   =>    |- C e. { A , B , C }
Theorem	snnzg 3809	The singleton of a set is not empty. It is also inhabited as shown at snmg 3810. (Contributed by NM, 14-Dec-2008.)
|- ( A e. V -> { A } =/= (/) )
Theorem	snmg 3810*	The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
|- ( A e. V -> E. x x e. { A } )
Theorem	snnz 3811	The singleton of a set is not empty. It is also inhabited as shown at snm 3812. (Contributed by NM, 10-Apr-1994.)
|- A e. _V   =>    |- { A } =/= (/)
Theorem	snm 3812*	The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
|- A e. _V   =>    |- E. x x e. { A }
Theorem	snmb 3813*	A singleton is inhabited iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.) (Revised by Jim Kingdon, 29-Dec-2025.)
|- ( A e. _V <-> E. x x e. { A } )
Theorem	prmg 3814*	A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
|- ( A e. V -> E. x x e. { A , B } )
Theorem	prnz 3815	A pair containing a set is not empty. It is also inhabited (see prm 3816). (Contributed by NM, 9-Apr-1994.)
|- A e. _V   =>    |- { A , B } =/= (/)
Theorem	prm 3816*	A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
|- A e. _V   =>    |- E. x x e. { A , B }
Theorem	prnzg 3817	A pair containing a set is not empty. It is also inhabited (see prmg 3814). (Contributed by FL, 19-Sep-2011.)
|- ( A e. V -> { A , B } =/= (/) )
Theorem	tpnz 3818	A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)
|- A e. _V   =>    |- { A , B , C } =/= (/)
Theorem	snssOLD 3819	Obsolete version of snss 3829 as of 1-Jan-2025. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
|- A e. _V   =>    |- ( A e. B <-> { A } C_ B )
Theorem	eldifsn 3820	Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)
|- ( A e. ( B \ { C } ) <-> ( A e. B /\ A =/= C ) )
Theorem	ssdifsn 3821	Subset of a set with an element removed. (Contributed by Emmett Weisz, 7-Jul-2021.) (Proof shortened by JJ, 31-May-2022.)
|- ( A C_ ( B \ { C } ) <-> ( A C_ B /\ -. C e. A ) )
Theorem	eldifsni 3822	Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)
|- ( A e. ( B \ { C } ) -> A =/= C )
Theorem	neldifsn 3823	A is not in ( B \ { A } ). (Contributed by David Moews, 1-May-2017.)
|- -. A e. ( B \ { A } )
Theorem	neldifsnd 3824	A is not in ( B \ { A } ). Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> -. A e. ( B \ { A } ) )
Theorem	rexdifsn 3825	Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)
|- ( E. x e. ( A \ { B } ) ph <-> E. x e. A ( x =/= B /\ ph ) )
Theorem	eldifvsn 3826	A set is an element of the universal class excluding a singleton iff it is not the singleton element. (Contributed by AV, 7-Apr-2019.)
|- ( A e. V -> ( A e. ( _V \ { B } ) <-> A =/= B ) )
Theorem	snssb 3827	Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025.)
|- ( { A } C_ B <-> ( A e. _V -> A e. B ) )
Theorem	snssg 3828	The singleton formed on a set is included in a class if and only if the set is an element of that class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.) (Proof shortened by BJ, 1-Jan-2025.)
|- ( A e. V -> ( A e. B <-> { A } C_ B ) )
Theorem	snss 3829	The singleton of an element of a class is a subset of the class (inference form of snssg 3828). Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 21-Jun-1993.) (Proof shortened by BJ, 1-Jan-2025.)
|- A e. _V   =>    |- ( A e. B <-> { A } C_ B )
Theorem	snssgOLD 3830	Obsolete version of snssgOLD 3830 as of 1-Jan-2025. (Contributed by NM, 22-Jul-2001.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( A e. V -> ( A e. B <-> { A } C_ B ) )
Theorem	difsn 3831	An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( -. A e. B -> ( B \ { A } ) = B )
Theorem	difprsnss 3832	Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( { A , B } \ { A } ) C_ { B }
Theorem	difprsn1 3833	Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)
|- ( A =/= B -> ( { A , B } \ { A } ) = { B } )
Theorem	difprsn2 3834	Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
|- ( A =/= B -> ( { A , B } \ { B } ) = { A } )
Theorem	diftpsn3 3835	Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
|- ( ( A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = { A , B } )
Theorem	difpr 3836	Removing two elements as pair of elements corresponds to removing each of the two elements as singletons. (Contributed by Alexander van der Vekens, 13-Jul-2018.)
|- ( A \ { B , C } ) = ( ( A \ { B } ) \ { C } )
Theorem	difsnb 3837	( B \ { A } ) equals B if and only if A is not a member of B. Generalization of difsn 3831. (Contributed by David Moews, 1-May-2017.)
|- ( -. A e. B <-> ( B \ { A } ) = B )
Theorem	snssi 3838	The singleton of an element of a class is a subset of the class. (Contributed by NM, 6-Jun-1994.)
|- ( A e. B -> { A } C_ B )
Theorem	snssd 3839	The singleton of an element of a class is a subset of the class (deduction form). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph -> A e. B )   =>    |- ( ph -> { A } C_ B )
Theorem	difsnss 3840	If we remove a single element from a class then put it back in, we end up with a subset of the original class. If equality is decidable, we can replace subset with equality as seen in nndifsnid 6740. (Contributed by Jim Kingdon, 10-Aug-2018.)
|- ( B e. A -> ( ( A \ { B } ) u. { B } ) C_ A )
Theorem	pw0 3841	Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ~P (/) = { (/) }
Theorem	snsspr1 3842	A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)
|- { A } C_ { A , B }
Theorem	snsspr2 3843	A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)
|- { B } C_ { A , B }
Theorem	snsstp1 3844	A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
|- { A } C_ { A , B , C }
Theorem	snsstp2 3845	A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
|- { B } C_ { A , B , C }
Theorem	snsstp3 3846	A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
|- { C } C_ { A , B , C }
Theorem	prsstp12 3847	A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)
|- { A , B } C_ { A , B , C }
Theorem	prsstp13 3848	A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)
|- { A , C } C_ { A , B , C }
Theorem	prsstp23 3849	A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)
|- { B , C } C_ { A , B , C }
Theorem	prss 3850	A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- A e. _V   &    |- B e. _V   =>    |- ( ( A e. C /\ B e. C ) <-> { A , B } C_ C )
Theorem	prssg 3851	A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( ( A e. V /\ B e. W ) -> ( ( A e. C /\ B e. C ) <-> { A , B } C_ C ) )
Theorem	prssi 3852	A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.)
|- ( ( A e. C /\ B e. C ) -> { A , B } C_ C )
Theorem	prssd 3853	Deduction version of prssi 3852: A pair of elements of a class is a subset of the class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A e. C )   &    |- ( ph -> B e. C )   =>    |- ( ph -> { A , B } C_ C )
Theorem	prsspwg 3854	An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.) (Revised by NM, 18-Jan-2018.)
|- ( ( A e. V /\ B e. W ) -> ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) ) )
Theorem	ssprss 3855	A pair as subset of a pair. (Contributed by AV, 26-Oct-2020.)
|- ( ( A e. V /\ B e. W ) -> ( { A , B } C_ { C , D } <-> ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) ) )
Theorem	ssprsseq 3856	A proper pair is a subset of a pair iff it is equal to the superset. (Contributed by AV, 26-Oct-2020.)
|- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } C_ { C , D } <-> { A , B } = { C , D } ) )
Theorem	sssnr 3857	Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4315. (Contributed by Jim Kingdon, 10-Aug-2018.)
|- ( ( A = (/) \/ A = { B } ) -> A C_ { B } )
Theorem	sssnm 3858*	The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
|- ( E. x x e. A -> ( A C_ { B } <-> A = { B } ) )
Theorem	eqsnm 3859*	Two ways to express that an inhabited set equals a singleton. (Contributed by Jim Kingdon, 11-Aug-2018.)
|- ( E. x x e. A -> ( A = { B } <-> A. x e. A x = B ) )
Theorem	ssprr 3860	The subsets of a pair. (Contributed by Jim Kingdon, 11-Aug-2018.)
|- ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) -> A C_ { B , C } )
Theorem	sstpr 3861	The subsets of a triple. (Contributed by Jim Kingdon, 11-Aug-2018.)
|- ( ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) \/ ( ( A = { D } \/ A = { B , D } ) \/ ( A = { C , D } \/ A = { B , C , D } ) ) ) -> A C_ { B , C , D } )
Theorem	tpss 3862	A triplet of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   =>    |- ( ( A e. D /\ B e. D /\ C e. D ) <-> { A , B , C } C_ D )
Theorem	tpssi 3863	A triple of elements of a class is a subset of the class. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
|- ( ( A e. D /\ B e. D /\ C e. D ) -> { A , B , C } C_ D )
Theorem	sneqr 3864	If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 27-Aug-1993.)
|- A e. _V   =>    |- ( { A } = { B } -> A = B )
Theorem	snsssn 3865	If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)
|- A e. _V   =>    |- ( { A } C_ { B } -> A = B )
Theorem	sneqrg 3866	Closed form of sneqr 3864. (Contributed by Scott Fenton, 1-Apr-2011.)
|- ( A e. V -> ( { A } = { B } -> A = B ) )
Theorem	sneqbg 3867	Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)
|- ( A e. V -> ( { A } = { B } <-> A = B ) )
Theorem	snsspw 3868	The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)
|- { A } C_ ~P A
Theorem	prsspw 3869	An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- A e. _V   &    |- B e. _V   =>    |- ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) )
Theorem	preqr1g 3870	Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3872. (Contributed by Jim Kingdon, 21-Sep-2018.)
|- ( ( A e. _V /\ B e. _V ) -> ( { A , C } = { B , C } -> A = B ) )
Theorem	preqr2g 3871	Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the second elements are equal. Closed form of preqr2 3873. (Contributed by Jim Kingdon, 21-Sep-2018.)
|- ( ( A e. _V /\ B e. _V ) -> ( { C , A } = { C , B } -> A = B ) )
Theorem	preqr1 3872	Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.)
|- A e. _V   &    |- B e. _V   =>    |- ( { A , C } = { B , C } -> A = B )
Theorem	preqr2 3873	Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 5-Aug-1993.)
|- A e. _V   &    |- B e. _V   =>    |- ( { C , A } = { C , B } -> A = B )
Theorem	preq12b 3874	Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
Theorem	prel12 3875	Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( -. A = B -> ( { A , B } = { C , D } <-> ( A e. { C , D } /\ B e. { C , D } ) ) )
Theorem	opthpr 3876	A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( A =/= D -> ( { A , B } = { C , D } <-> ( A = C /\ B = D ) ) )
Theorem	preq12bg 3877	Closed form of preq12b 3874. (Contributed by Scott Fenton, 28-Mar-2014.)
|- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) -> ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) )
Theorem	prneimg 3878	Two pairs are not equal if at least one element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
|- ( ( ( A e. U /\ B e. V ) /\ ( C e. X /\ D e. Y ) ) -> ( ( ( A =/= C /\ A =/= D ) \/ ( B =/= C /\ B =/= D ) ) -> { A , B } =/= { C , D } ) )
Theorem	preqsn 3879	Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   =>    |- ( { A , B } = { C } <-> ( A = B /\ B = C ) )
Theorem	elpr2elpr 3880*	For an element A of an unordered pair which is a subset of a given set V, there is another (maybe the same) element b of the given set V being an element of the unordered pair. (Contributed by AV, 5-Dec-2020.)
|- ( ( X e. V /\ Y e. V /\ A e. { X , Y } ) -> E. b e. V { X , Y } = { A , b } )
Theorem	dfopg 3881	Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( ( A e. V /\ B e. W ) -> <. A , B >. = { { A } , { A , B } } )
Theorem	dfop 3882	Value of an ordered pair when the arguments are sets, with the conclusion corresponding to Kuratowski's original definition. (Contributed by NM, 25-Jun-1998.)
|- A e. _V   &    |- B e. _V   =>    |- <. A , B >. = { { A } , { A , B } }
Theorem	opeq1 3883	Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( A = B -> <. A , C >. = <. B , C >. )
Theorem	opeq2 3884	Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( A = B -> <. C , A >. = <. C , B >. )
Theorem	opeq12 3885	Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
|- ( ( A = C /\ B = D ) -> <. A , B >. = <. C , D >. )
Theorem	opeq1i 3886	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
|- A = B   =>    |- <. A , C >. = <. B , C >.
Theorem	opeq2i 3887	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
|- A = B   =>    |- <. C , A >. = <. C , B >.
Theorem	opeq12i 3888	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
|- A = B   &    |- C = D   =>    |- <. A , C >. = <. B , D >.
Theorem	opeq1d 3889	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> <. A , C >. = <. B , C >. )
Theorem	opeq2d 3890	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> <. C , A >. = <. C , B >. )
Theorem	opeq12d 3891	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> <. A , C >. = <. B , D >. )
Theorem	oteq1 3892	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
|- ( A = B -> <. A , C , D >. = <. B , C , D >. )
Theorem	oteq2 3893	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
|- ( A = B -> <. C , A , D >. = <. C , B , D >. )
Theorem	oteq3 3894	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
|- ( A = B -> <. C , D , A >. = <. C , D , B >. )
Theorem	oteq1d 3895	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- ( ph -> A = B )   =>    |- ( ph -> <. A , C , D >. = <. B , C , D >. )
Theorem	oteq2d 3896	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- ( ph -> A = B )   =>    |- ( ph -> <. C , A , D >. = <. C , B , D >. )
Theorem	oteq3d 3897	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- ( ph -> A = B )   =>    |- ( ph -> <. C , D , A >. = <. C , D , B >. )
Theorem	oteq123d 3898	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   &    |- ( ph -> E = F )   =>    |- ( ph -> <. A , C , E >. = <. B , D , F >. )
Theorem	nfop 3899	Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x <. A , B >.
Theorem	nfopd 3900	Deduction version of bound-variable hypothesis builder nfop 3899. This shows how the deduction version of a not-free theorem such as nfop 3899 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)
|- ( ph -> F/_ x A )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> F/_ x <. A , B >. )