P. 33 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3201-3300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sbcnestg 3201*	Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
|- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )
Theorem	csbnestg 3202*	Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
|- ( A e. V -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )
Theorem	csbnest1g 3203	Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
|- ( A e. V -> [_ A / x ]_ [_ B / x ]_ C = [_ [_ A / x ]_ B / x ]_ C )
Theorem	csbidmg 3204*	Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)
|- ( A e. V -> [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B )
Theorem	sbcco3g 3205*	Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
|- ( x = A -> B = C )   =>    |- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. C / y ]. ph ) )
Theorem	csbco3g 3206*	Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
|- ( x = A -> B = C )   =>    |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ D = [_ C / y ]_ D )
Theorem	rspcsbela 3207*	Special case related to rspsbc 3135. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
|- ( ( A e. B /\ A. x e. B C e. D ) -> [_ A / x ]_ C e. D )
Theorem	sbnfc2 3208*	Two ways of expressing " x is (effectively) not free in A." (Contributed by Mario Carneiro, 14-Oct-2016.)
|- ( F/_ x A <-> A. y A. z [_ y / x ]_ A = [_ z / x ]_ A )
Theorem	csbabg 3209*	Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( A e. V -> [_ A / x ]_ { y | ph } = { y | [. A / x ]. ph } )
Theorem	cbvralcsf 3210	A more general version of cbvralf 2777 that doesn't require A and B to be distinct from x or y. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
|- F/_ y A   &    |- F/_ x B   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> A = B )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x e. A ph <-> A. y e. B ps )
Theorem	cbvrexcsf 3211	A more general version of cbvrexf 2778 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.)
|- F/_ y A   &    |- F/_ x B   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> A = B )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x e. A ph <-> E. y e. B ps )
Theorem	cbvreucsf 3212	A more general version of cbvreuv 2788 that has no distinct variable rextrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
|- F/_ y A   &    |- F/_ x B   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> A = B )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph <-> E! y e. B ps )
Theorem	cbvrabcsf 3213	A more general version of cbvrab 2819 with no distinct variable restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)
|- F/_ y A   &    |- F/_ x B   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> A = B )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- { x e. A | ph } = { y e. B | ps }
Theorem	cbvralv2 3214*	Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)
|- ( x = y -> ( ps <-> ch ) )   &    |- ( x = y -> A = B )   =>    |- ( A. x e. A ps <-> A. y e. B ch )
Theorem	cbvrexv2 3215*	Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)
|- ( x = y -> ( ps <-> ch ) )   &    |- ( x = y -> A = B )   =>    |- ( E. x e. A ps <-> E. y e. B ch )
Theorem	rspc2vd 3216*	Deduction version of 2-variable restricted specialization, using implicit substitution. Notice that the class D for the second set variable y may depend on the first set variable x. (Contributed by AV, 29-Mar-2021.)
|- ( x = A -> ( th <-> ch ) )   &    |- ( y = B -> ( ch <-> ps ) )   &    |- ( ph -> A e. C )   &    |- ( ( ph /\ x = A ) -> D = E )   &    |- ( ph -> B e. E )   =>    |- ( ph -> ( A. x e. C A. y e. D th -> ps ) )
2.1.11  Define basic set operations and relations
Syntax	cdif 3217	Extend class notation to include class difference (read: " A minus B").
class ( A \ B )
Syntax	cun 3218	Extend class notation to include union of two classes (read: " A union B").
class ( A u. B )
Syntax	cin 3219	Extend class notation to include the intersection of two classes (read: " A intersect B").
class ( A i^i B )
Syntax	wss 3220	Extend wff notation to include the subclass relation. This is read " A is a subclass of B " or " B includes A". When A exists as a set, it is also read " A is a subset of B".
wff A C_ B
Theorem	difjust 3221*	Soundness justification theorem for df-dif 3222. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- { x | ( x e. A /\ -. x e. B ) } = { y | ( y e. A /\ -. y e. B ) }
Definition	df-dif 3222*	Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Contrast this operation with union ( A u. B ) (df-un 3224) and intersection ( A i^i B ) (df-in 3226). Several notations are used in the literature; we chose the \ convention used in Definition 5.3 of [Eisenberg] p. 67 instead of the more common minus sign to reserve the latter for later use in, e.g., arithmetic. We will use the terminology " A excludes B " to mean A \ B. We will use " B is removed from A " to mean A \ { B } i.e. the removal of an element or equivalently the exclusion of a singleton. (Contributed by NM, 29-Apr-1994.)
|- ( A \ B ) = { x | ( x e. A /\ -. x e. B ) }
Theorem	unjust 3223*	Soundness justification theorem for df-un 3224. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- { x | ( x e. A \/ x e. B ) } = { y | ( y e. A \/ y e. B ) }
Definition	df-un 3224*	Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with difference ( A \ B ) (df-dif 3222) and intersection ( A i^i B ) (df-in 3226). (Contributed by NM, 23-Aug-1993.)
|- ( A u. B ) = { x | ( x e. A \/ x e. B ) }
Theorem	injust 3225*	Soundness justification theorem for df-in 3226. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- { x | ( x e. A /\ x e. B ) } = { y | ( y e. A /\ y e. B ) }
Definition	df-in 3226*	Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with union ( A u. B ) (df-un 3224) and difference ( A \ B ) (df-dif 3222). (Contributed by NM, 29-Apr-1994.)
|- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
Theorem	dfin5 3227*	Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)
|- ( A i^i B ) = { x e. A | x e. B }
Theorem	dfdif2 3228*	Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
|- ( A \ B ) = { x e. A | -. x e. B }
Theorem	eldif 3229	Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
|- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )
Theorem	eldifd 3230	If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3229. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. B )   &    |- ( ph -> -. A e. C )   =>    |- ( ph -> A e. ( B \ C ) )
Theorem	eldifad 3231	If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3229. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. ( B \ C ) )   =>    |- ( ph -> A e. B )
Theorem	eldifbd 3232	If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3229. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. ( B \ C ) )   =>    |- ( ph -> -. A e. C )
2.1.12  Subclasses and subsets
Definition	df-ss 3233	Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. Note that A C_ A (proved in ssid 3268). For a more traditional definition, but requiring a dummy variable, see ssalel 3235. Other possible definitions are given by dfss3 3236, ssequn1 3399, ssequn2 3402, and sseqin2 3450. (Contributed by NM, 27-Apr-1994.)
|- ( A C_ B <-> ( A i^i B ) = A )
Theorem	dfss 3234	Variant of subclass definition df-ss 3233. (Contributed by NM, 3-Sep-2004.)
|- ( A C_ B <-> A = ( A i^i B ) )
Theorem	ssalel 3235*	Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.)
|- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
Theorem	dfss3 3236*	Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
|- ( A C_ B <-> A. x e. A x e. B )
Theorem	dfss2 3237	Alternate definition of the subclass relationship between two classes. Exercise 9 of [TakeutiZaring] p. 18. This is another name for df-ss 3233 which is more consistent with the naming in the Metamath Proof Explorer. (Contributed by NM, 27-Apr-1994.)
|- ( A C_ B <-> ( A i^i B ) = A )
Theorem	dfssf 3238	Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
Theorem	dfss2f 3239	Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
Theorem	dfss3f 3240	Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A C_ B <-> A. x e. A x e. B )
Theorem	nfss 3241	If x is not free in A and B, it is not free in A C_ B. (Contributed by NM, 27-Dec-1996.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/ x A C_ B
Theorem	ssel 3242	Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
|- ( A C_ B -> ( C e. A -> C e. B ) )
Theorem	ssel2 3243	Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
|- ( ( A C_ B /\ C e. A ) -> C e. B )
Theorem	sseli 3244	Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
|- A C_ B   =>    |- ( C e. A -> C e. B )
Theorem	sselii 3245	Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
|- A C_ B   &    |- C e. A   =>    |- C e. B
Theorem	sselid 3246	Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
|- A C_ B   &    |- ( ph -> C e. A )   =>    |- ( ph -> C e. B )
Theorem	sseld 3247	Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
|- ( ph -> A C_ B )   =>    |- ( ph -> ( C e. A -> C e. B ) )
Theorem	sselda 3248	Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
|- ( ph -> A C_ B )   =>    |- ( ( ph /\ C e. A ) -> C e. B )
Theorem	sseldd 3249	Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
|- ( ph -> A C_ B )   &    |- ( ph -> C e. A )   =>    |- ( ph -> C e. B )
Theorem	ssneld 3250	If a class is not in another class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A C_ B )   =>    |- ( ph -> ( -. C e. B -> -. C e. A ) )
Theorem	ssneldd 3251	If an element is not in a class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A C_ B )   &    |- ( ph -> -. C e. B )   =>    |- ( ph -> -. C e. A )
Theorem	ssriv 3252*	Inference based on subclass definition. (Contributed by NM, 5-Aug-1993.)
|- ( x e. A -> x e. B )   =>    |- A C_ B
Theorem	ssrd 3253	Deduction based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
|- F/ x ph   &    |- F/_ x A   &    |- F/_ x B   &    |- ( ph -> ( x e. A -> x e. B ) )   =>    |- ( ph -> A C_ B )
Theorem	ssrdv 3254*	Deduction based on subclass definition. (Contributed by NM, 15-Nov-1995.)
|- ( ph -> ( x e. A -> x e. B ) )   =>    |- ( ph -> A C_ B )
Theorem	sstr2 3255	Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( A C_ B -> ( B C_ C -> A C_ C ) )
Theorem	sstr 3256	Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
|- ( ( A C_ B /\ B C_ C ) -> A C_ C )
Theorem	sstri 3257	Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
|- A C_ B   &    |- B C_ C   =>    |- A C_ C
Theorem	sstrd 3258	Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
|- ( ph -> A C_ B )   &    |- ( ph -> B C_ C )   =>    |- ( ph -> A C_ C )
Theorem	sstrid 3259	Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
|- A C_ B   &    |- ( ph -> B C_ C )   =>    |- ( ph -> A C_ C )
Theorem	sstrdi 3260	Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph -> A C_ B )   &    |- B C_ C   =>    |- ( ph -> A C_ C )
Theorem	sylan9ss 3261	A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( ph -> A C_ B )   &    |- ( ps -> B C_ C )   =>    |- ( ( ph /\ ps ) -> A C_ C )
Theorem	sylan9ssr 3262	A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
|- ( ph -> A C_ B )   &    |- ( ps -> B C_ C )   =>    |- ( ( ps /\ ph ) -> A C_ C )
Theorem	eqss 3263	The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 5-Aug-1993.)
|- ( A = B <-> ( A C_ B /\ B C_ A ) )
Theorem	eqssi 3264	Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)
|- A C_ B   &    |- B C_ A   =>    |- A = B
Theorem	eqssd 3265	Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)
|- ( ph -> A C_ B )   &    |- ( ph -> B C_ A )   =>    |- ( ph -> A = B )
Theorem	eqrd 3266	Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)
|- F/ x ph   &    |- F/_ x A   &    |- F/_ x B   &    |- ( ph -> ( x e. A <-> x e. B ) )   =>    |- ( ph -> A = B )
Theorem	eqelssd 3267*	Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.)
|- ( ph -> A C_ B )   &    |- ( ( ph /\ x e. B ) -> x e. A )   =>    |- ( ph -> A = B )
Theorem	ssid 3268	Any class is a subclass of itself. Exercise 10 of [TakeutiZaring] p. 18. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- A C_ A
Theorem	ssidd 3269	Weakening of ssid 3268. (Contributed by BJ, 1-Sep-2022.)
|- ( ph -> A C_ A )
Theorem	ssv 3270	Any class is a subclass of the universal class. Dual of 0ss 3561. (Contributed by NM, 31-Oct-1995.)
|- A C_ _V
Theorem	sseq1 3271	Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
|- ( A = B -> ( A C_ C <-> B C_ C ) )
Theorem	sseq2 3272	Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
|- ( A = B -> ( C C_ A <-> C C_ B ) )
Theorem	sseq12 3273	Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
|- ( ( A = B /\ C = D ) -> ( A C_ C <-> B C_ D ) )
Theorem	sseq1i 3274	An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
|- A = B   =>    |- ( A C_ C <-> B C_ C )
Theorem	sseq2i 3275	An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
|- A = B   =>    |- ( C C_ A <-> C C_ B )
Theorem	sseq12i 3276	An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|- A = B   &    |- C = D   =>    |- ( A C_ C <-> B C_ D )
Theorem	sseq1d 3277	An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A C_ C <-> B C_ C ) )
Theorem	sseq2d 3278	An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C C_ A <-> C C_ B ) )
Theorem	sseq12d 3279	An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A C_ C <-> B C_ D ) )
Theorem	eqsstri 3280	Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
|- A = B   &    |- B C_ C   =>    |- A C_ C
Theorem	eqsstrri 3281	Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
|- B = A   &    |- B C_ C   =>    |- A C_ C
Theorem	sseqtri 3282	Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
|- A C_ B   &    |- B = C   =>    |- A C_ C
Theorem	sseqtrri 3283	Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
|- A C_ B   &    |- C = B   =>    |- A C_ C
Theorem	eqsstrd 3284	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
|- ( ph -> A = B )   &    |- ( ph -> B C_ C )   =>    |- ( ph -> A C_ C )
Theorem	eqsstrrd 3285	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
|- ( ph -> B = A )   &    |- ( ph -> B C_ C )   =>    |- ( ph -> A C_ C )
Theorem	sseqtrd 3286	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
|- ( ph -> A C_ B )   &    |- ( ph -> B = C )   =>    |- ( ph -> A C_ C )
Theorem	sseqtrrd 3287	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
|- ( ph -> A C_ B )   &    |- ( ph -> C = B )   =>    |- ( ph -> A C_ C )
Theorem	3sstr3i 3288	Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|- A C_ B   &    |- A = C   &    |- B = D   =>    |- C C_ D
Theorem	3sstr4i 3289	Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|- A C_ B   &    |- C = A   &    |- D = B   =>    |- C C_ D
Theorem	3sstr3g 3290	Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
|- ( ph -> A C_ B )   &    |- A = C   &    |- B = D   =>    |- ( ph -> C C_ D )
Theorem	3sstr4g 3291	Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|- ( ph -> A C_ B )   &    |- C = A   &    |- D = B   =>    |- ( ph -> C C_ D )
Theorem	3sstr3d 3292	Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
|- ( ph -> A C_ B )   &    |- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> C C_ D )
Theorem	3sstr4d 3293	Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|- ( ph -> A C_ B )   &    |- ( ph -> C = A )   &    |- ( ph -> D = B )   =>    |- ( ph -> C C_ D )
Theorem	eqsstrid 3294	B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
|- A = B   &    |- ( ph -> B C_ C )   =>    |- ( ph -> A C_ C )
Theorem	eqsstrrid 3295	B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
|- B = A   &    |- ( ph -> B C_ C )   =>    |- ( ph -> A C_ C )
Theorem	sseqtrdi 3296	A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
|- ( ph -> A C_ B )   &    |- B = C   =>    |- ( ph -> A C_ C )
Theorem	sseqtrrdi 3297	A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
|- ( ph -> A C_ B )   &    |- C = B   =>    |- ( ph -> A C_ C )
Theorem	sseqtrid 3298	Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- B C_ A   &    |- ( ph -> A = C )   =>    |- ( ph -> B C_ C )
Theorem	sseqtrrid 3299	Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- B C_ A   &    |- ( ph -> C = A )   =>    |- ( ph -> B C_ C )
Theorem	eqsstrdi 3300	A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- ( ph -> A = B )   &    |- B C_ C   =>    |- ( ph -> A C_ C )