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Theorem List for Metamath Proof Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem0iin 3901 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
 |-  |^|_ x  e.  (/)  A  =  _V
 
Theoremviin 3902* Indexed intersection with a universal index class. When  A doesn't depend on  x, this evaluates to  A by 19.3 1760 and abid2 2373. When  A  =  x, this evaluates to  (/) by intiin 3897 and intv 4124. (Contributed by NM, 11-Sep-2008.)
 |-  |^|_ x  e.  _V  A  =  { y  |  A. x  y  e.  A }
 
Theoremiunn0 3903* There is a non-empty class in an indexed collection  B ( x ) iff the indexed union of them is non-empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( E. x  e.  A  B  =/=  (/)  <->  U_ x  e.  A  B  =/=  (/) )
 
Theoremiinab 3904* Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
 |-  |^|_ x  e.  A  {
 y  |  ph }  =  { y  |  A. x  e.  A  ph }
 
Theoremiinrab 3905* Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)
 |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  { y  e.  B  |  ph }  =  { y  e.  B  |  A. x  e.  A  ph
 } )
 
Theoremiinrab2 3906* Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)
 |-  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  { y  e.  B  |  A. x  e.  A  ph }
 
Theoremiunin2 3907* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3896 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 )
 
Theoremiunin1 3908* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3896 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 )
 
Theoremiinun2 3909* Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3897 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
 |-  |^|_ x  e.  A  ( B  u.  C )  =  ( B  u.  |^|_
 x  e.  A  C )
 
Theoremiundif2 3910* Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 3897 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
 |-  U_ x  e.  A  ( B  \  C )  =  ( B  \  |^|_
 x  e.  A  C )
 
Theorem2iunin 3911* 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 )
 
Theoremiindif2 3912* Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 3896 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)
 |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( B  \  C )  =  ( B  \  U_ x  e.  A  C ) )
 
Theoremiinin2 3913* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3897 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  |^|_ x  e.  A  C ) )
 
Theoremiinin1 3914* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3897 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( C  i^i  B )  =  ( |^|_ x  e.  A  C  i^i  B ) )
 
Theoremelriin 3915* 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 ) )
 
Theoremriin0 3916* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  A )
 
Theoremriinn0 3917* Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( A. x  e.  X  S  C_  A  /\  X  =/=  (/) )  ->  ( A  i^i  |^|_ x  e.  X  S )  = 
 |^|_ x  e.  X  S )
 
Theoremriinrab 3918* Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  { y  e.  A  |  A. x  e.  X  ph
 }
 
Theoremiinxsng 3919* 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 )
 
Theoremiinxprg 3920* 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 ) )
 
Theoremiunxsng 3921* 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 )
 
Theoremiunxsn 3922* 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
 
Theoremiunun 3923 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 )
 
Theoremiunxun 3924 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 )
 
Theoremiunxiun 3925* 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
 
Theoremiinuni 3926* A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
 |-  ( A  u.  |^| B )  =  |^|_ x  e.  B  ( A  u.  x )
 
Theoremiununi 3927* A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
 |-  ( ( B  =  (/) 
 ->  A  =  (/) )  <->  ( A  u.  U. B )  =  U_ x  e.  B  ( A  u.  x ) )
 
Theoremsspwuni 3928 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
 |-  ( A  C_  ~P B  <->  U. A  C_  B )
 
Theorempwssb 3929* 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 )
 
Theoremelpwuni 3930 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
 |-  ( B  e.  A  ->  ( A  C_  ~P B  <->  U. A  =  B ) )
 
Theoremiinpw 3931* 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
 
Theoremiunpwss 3932* 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
 
Theoremrintn0 3933 Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)
 |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  ->  ( A  i^i  |^| X )  = 
 |^| X )
 
2.1.21  Disjointness
 
Syntaxwdisj 3934 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
 
Definitiondf-disj 3935* 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.)
 |-  (Disj  x  e.  A B 
 <-> 
 A. y E* x ( x  e.  A  /\  y  e.  B ) )
 
Theoremdisjss2 3936 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 ) )
 
Theoremdisjeq2 3937 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 ) )
 
Theoremdisjeq2dv 3938* 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 ) )
 
Theoremdisjss1 3939* 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 ) )
 
Theoremdisjeq1 3940* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( A  =  B  ->  (Disj  x  e.  A C 
 <-> Disj  x  e.  B C ) )
 
Theoremdisjeq1d 3941* 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 ) )
 
Theoremdisjeq12d 3942* 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 ) )
 
Theoremcbvdisj 3943* 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 )
 
Theoremcbvdisjv 3944* 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 )
 
Theoremnfdisj 3945 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  F/_ y A   &    |-  F/_ y B   =>    |-  F/ yDisj  x  e.  A B
 
Theoremnfdisj1 3946 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- 
 F/ xDisj  x  e.  A B
 
Theoremdisjor 3947* Two ways to say that a collection 
B ( i ) for  i  e.  A is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
 |-  ( i  =  j 
 ->  B  =  C )   =>    |-  (Disj  i  e.  A B  <->  A. i  e.  A  A. j  e.  A  (
 i  =  j  \/  ( B  i^i  C )  =  (/) ) )
 
TheoremdisjmoOLD 3948* Two ways to say that a collection 
B ( i ) for  i  e.  A is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (New usage is discouraged.)
 |-  ( i  =  j 
 ->  B  =  C )   =>    |-  ( A. x E* i
 ( i  e.  A  /\  x  e.  B ) 
 <-> 
 A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( B  i^i  C )  =  (/) ) )
 
Theoremdisjors 3949* Two ways to say that a collection 
B ( i ) for  i  e.  A is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  (Disj  x  e.  A B 
 <-> 
 A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
 
Theoremdisji2 3950* Property of a disjoint collection: if  B ( X )  =  C and  B ( Y )  =  D, and  X  =/=  Y, then  C and  D are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( x  =  X  ->  B  =  C )   &    |-  ( x  =  Y  ->  B  =  D )   =>    |-  ( (Disj  x  e.  A B  /\  ( X  e.  A  /\  Y  e.  A )  /\  X  =/=  Y )  ->  ( C  i^i  D )  =  (/) )
 
Theoremdisji 3951* Property of a disjoint collection: if  B ( X )  =  C and  B ( Y )  =  D have a common element  Z, then  X  =  Y. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( x  =  X  ->  B  =  C )   &    |-  ( x  =  Y  ->  B  =  D )   =>    |-  ( (Disj  x  e.  A B  /\  ( X  e.  A  /\  Y  e.  A )  /\  ( Z  e.  C  /\  Z  e.  D ) )  ->  X  =  Y )
 
Theoreminvdisj 3952* 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 )
 
Theoremdisjiun 3953* A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
 |-  ( (Disj  x  e.  A B  /\  ( C 
 C_  A  /\  D  C_  A  /\  ( C  i^i  D )  =  (/) ) )  ->  ( U_ x  e.  C  B  i^i  U_ x  e.  D  B )  =  (/) )
 
TheoremdisjiunOLD 3954* A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (New usage is discouraged.)
 |-  ( ( A. y E* x ( x  e.  A  /\  y  e.  B )  /\  ( C  C_  A  /\  D  C_  A  /\  ( C  i^i  D )  =  (/) ) )  ->  ( U_ x  e.  C  B  i^i  U_ x  e.  D  B )  =  (/) )
 
Theoremsndisj 3955 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  A { x }
 
Theorem0disj 3956 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  A (/)
 
Theoremdisjxsn 3957* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  { A } B
 
Theoremdisjx0 3958 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  (/) B
 
Theoremdisjprg 3959* A pair collection is disjoint iff the two sets in the family have empty intersection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( x  =  A  ->  C  =  D )   &    |-  ( x  =  B  ->  C  =  E )   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B ) 
 ->  (Disj  x  e.  { A ,  B } C  <->  ( D  i^i  E )  =  (/) ) )
 
Theoremdisjxiun 3960* An indexed union of a disjoint collection of disjoint collections is disjoint if each component is disjoint, and the disjoint unions in the collection are also disjoint. Note that  B ( y ) and  C
( x ) may have the displayed free variables. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  (Disj  y  e.  A B  ->  (Disj  x  e.  U_ y  e.  A  B C  <->  ( A. y  e.  A Disj  x  e.  B C  /\ Disj  y  e.  A U_ x  e.  B  C ) ) )
 
Theoremdisjxun 3961* The union of two disjoint collections. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( x  =  y 
 ->  C  =  D )   =>    |-  ( ( A  i^i  B )  =  (/)  ->  (Disj  x  e.  ( A  u.  B ) C  <->  (Disj  x  e.  A C  /\ Disj  x  e.  B C  /\  A. x  e.  A  A. y  e.  B  ( C  i^i  D )  =  (/) ) ) )
 
Theoremdisjss3 3962* Expand a disjoint collection with any number of empty sets. (Contributed by Mario Carneiro, 15-Nov-2016.)
 |-  ( ( A  C_  B  /\  A. x  e.  ( B  \  A ) C  =  (/) )  ->  (Disj  x  e.  A C  <-> Disj  x  e.  B C ) )
 
2.1.22  Binary relations
 
Syntaxwbr 3963 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. (For an example of how syntax could become ambiguous if we are not careful, see the comment in cneg 8971.)
 wff  A R B
 
Definitiondf-br 3964 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. Class  R often denotes a relation such as " < " that compares two classes  A and 
B, which might be numbers such as  1 and  2 (see df-ltxr 8805 for the specific definition of  <). As a wff, relations are true or false. For example,  ( R  =  { <. 2 ,  6
>. ,  <. 3 ,  9 >. }  ->  3 R 9 ) (ex-br 20726). Often class  R meets the  Rel criteria to be defined in df-rel 4641, and in particular  R may be a function (see df-fun 4648). 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. (Contributed by NM, 31-Dec-1993.)
 |-  ( A R B  <->  <. A ,  B >.  e.  R )
 
Theorembreq 3965 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
 |-  ( R  =  S  ->  ( A R B  <->  A S B ) )
 
Theorembreq1 3966 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
 |-  ( A  =  B  ->  ( A R C  <->  B R C ) )
 
Theorembreq2 3967 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
 |-  ( A  =  B  ->  ( C R A  <->  C R B ) )
 
Theorembreq12 3968 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A R C 
 <->  B R D ) )
 
Theorembreqi 3969 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
 |-  R  =  S   =>    |-  ( A R B 
 <->  A S B )
 
Theorembreq1i 3970 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  A  =  B   =>    |-  ( A R C 
 <->  B R C )
 
Theorembreq2i 3971 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  A  =  B   =>    |-  ( C R A 
 <->  C R B )
 
Theorembreq12i 3972 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 )
 
Theorembreq1d 3973 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A R C  <->  B R C ) )
 
Theorembreqd 3974 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C A D  <->  C B D ) )
 
Theorembreq2d 3975 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C R A  <->  C R B ) )
 
Theorembreq12d 3976 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 ) )
 
Theorembreq123d 3977 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 ) )
 
Theorembreqan12d 3978 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 ) )
 
Theorembreqan12rd 3979 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  C  =  D )   =>    |-  ( ( ps 
 /\  ph )  ->  ( A R C  <->  B R D ) )
 
Theoremnbrne1 3980 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
 |-  ( ( A R B  /\  -.  A R C )  ->  B  =/=  C )
 
Theoremnbrne2 3981 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
 |-  ( ( A R C  /\  -.  B R C )  ->  A  =/=  B )
 
Theoremeqbrtri 3982 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  B R C   =>    |-  A R C
 
Theoremeqbrtrd 3983 Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theoremeqbrtrri 3984 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A  =  B   &    |-  A R C   =>    |-  B R C
 
Theoremeqbrtrrd 3985 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A R C )   =>    |-  ( ph  ->  B R C )
 
Theorembreqtri 3986 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A R B   &    |-  B  =  C   =>    |-  A R C
 
Theorembreqtrd 3987 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
 |-  ( ph  ->  A R B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A R C )
 
Theorembreqtrri 3988 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 |-  A R B   &    |-  C  =  B   =>    |-  A R C
 
Theorembreqtrrd 3989 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
 |-  ( ph  ->  A R B )   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A R C )
 
Theorem3brtr3i 3990 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
 |-  A R B   &    |-  A  =  C   &    |-  B  =  D   =>    |-  C R D
 
Theorem3brtr4i 3991 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
 |-  A R B   &    |-  C  =  A   &    |-  D  =  B   =>    |-  C R D
 
Theorem3brtr3d 3992 Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
 |-  ( ph  ->  A R B )   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  C R D )
 
Theorem3brtr4d 3993 Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
 |-  ( ph  ->  A R B )   &    |-  ( ph  ->  C  =  A )   &    |-  ( ph  ->  D  =  B )   =>    |-  ( ph  ->  C R D )
 
Theorem3brtr3g 3994 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
 |-  ( ph  ->  A R B )   &    |-  A  =  C   &    |-  B  =  D   =>    |-  ( ph  ->  C R D )
 
Theorem3brtr4g 3995 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
 |-  ( ph  ->  A R B )   &    |-  C  =  A   &    |-  D  =  B   =>    |-  ( ph  ->  C R D )
 
Theoremsyl5eqbr 3996 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
 |-  A  =  B   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theoremsyl5eqbrr 3997 B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
 |-  B  =  A   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theoremsyl5breq 3998 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
 |-  A R B   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A R C )
 
Theoremsyl5breqr 3999 B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
 |-  A R B   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A R C )
 
Theoremsyl6eqbr 4000 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
 |-  ( ph  ->  A  =  B )   &    |-  B R C   =>    |-  ( ph  ->  A R C )
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