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Theorem List for Intuitionistic Logic Explorer - 2901-3000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcsbnest1g 2901 Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
(𝐴𝑉𝐴 / 𝑥𝐵 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶)
 
Theoremcsbidmg 2902* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)
(𝐴𝑉𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
 
Theoremsbcco3g 2903* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
 
Theoremcsbco3g 2904* Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐷)
 
Theoremrspcsbela 2905* Special case related to rspsbc 2840. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
((𝐴𝐵 ∧ ∀𝑥𝐵 𝐶𝐷) → 𝐴 / 𝑥𝐶𝐷)
 
Theoremsbnfc2 2906* Two ways of expressing "𝑥 is (effectively) not free in 𝐴." (Contributed by Mario Carneiro, 14-Oct-2016.)
(𝑥𝐴 ↔ ∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
 
Theoremcsbabg 2907* Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝐴𝑉𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑})
 
Theoremcbvralcsf 2908 A more general version of cbvralf 2527 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.)
𝑦𝐴    &   𝑥𝐵    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐵 𝜓)
 
Theoremcbvrexcsf 2909 A more general version of cbvrexf 2528 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.)
𝑦𝐴    &   𝑥𝐵    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐵 𝜓)
 
Theoremcbvreucsf 2910 A more general version of cbvreuv 2535 that has no distinct variable rextrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
𝑦𝐴    &   𝑥𝐵    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐵 𝜓)
 
Theoremcbvrabcsf 2911 A more general version of cbvrab 2555 with no distinct variable restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)
𝑦𝐴    &   𝑥𝐵    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑦𝐵𝜓}
 
Theoremcbvralv2 2912* 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 → (𝜓𝜒))    &   (𝑥 = 𝑦𝐴 = 𝐵)       (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐵 𝜒)
 
Theoremcbvrexv2 2913* 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 → (𝜓𝜒))    &   (𝑥 = 𝑦𝐴 = 𝐵)       (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒)
 
2.1.11  Define basic set operations and relations
 
Syntaxcdif 2914 Extend class notation to include class difference (read: "𝐴 minus 𝐵").
class (𝐴𝐵)
 
Syntaxcun 2915 Extend class notation to include union of two classes (read: "𝐴 union 𝐵").
class (𝐴𝐵)
 
Syntaxcin 2916 Extend class notation to include the intersection of two classes (read: "𝐴 intersect 𝐵").
class (𝐴𝐵)
 
Syntaxwss 2917 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 𝐴𝐵
 
Syntaxwpss 2918 Extend wff notation with proper subclass relation.
wff 𝐴𝐵
 
Theoremdifjust 2919* Soundness justification theorem for df-dif 2920. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴 ∧ ¬ 𝑦𝐵)}
 
Definitiondf-dif 2920* Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Contrast this operation with union (𝐴𝐵) (df-un 2922) and intersection (𝐴𝐵) (df-in 2924). 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 𝐴𝐵. 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.)
(𝐴𝐵) = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
 
Theoremunjust 2921* Soundness justification theorem for df-un 2922. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
 
Definitiondf-un 2922* Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with difference (𝐴𝐵) (df-dif 2920) and intersection (𝐴𝐵) (df-in 2924). (Contributed by NM, 23-Aug-1993.)
(𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
 
Theoreminjust 2923* Soundness justification theorem for df-in 2924. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
 
Definitiondf-in 2924* Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with union (𝐴𝐵) (df-un 2922) and difference (𝐴𝐵) (df-dif 2920). (Contributed by NM, 29-Apr-1994.)
(𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
 
Theoremdfin5 2925* Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)
(𝐴𝐵) = {𝑥𝐴𝑥𝐵}
 
Theoremdfdif2 2926* Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
(𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
 
Theoremeldif 2927 Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
(𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
 
Theoremeldifd 2928 If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 2927. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑 → ¬ 𝐴𝐶)       (𝜑𝐴 ∈ (𝐵𝐶))
 
Theoremeldifad 2929 If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 2927. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (𝐵𝐶))       (𝜑𝐴𝐵)
 
Theoremeldifbd 2930 If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 2927. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (𝐵𝐶))       (𝜑 → ¬ 𝐴𝐶)
 
2.1.12  Subclasses and subsets
 
Definitiondf-ss 2931 Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. Note that 𝐴𝐴 (proved in ssid 2964). Contrast this relationship with the relationship 𝐴𝐵 (as will be defined in df-pss 2933). For a more traditional definition, but requiring a dummy variable, see dfss2 2934 (or dfss3 2935 which is similar). (Contributed by NM, 27-Apr-1994.)
(𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
 
Theoremdfss 2932 Variant of subclass definition df-ss 2931. (Contributed by NM, 3-Sep-2004.)
(𝐴𝐵𝐴 = (𝐴𝐵))
 
Definitiondf-pss 2933 Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. Note that ¬ 𝐴𝐴 (proved in pssirr 3044). Contrast this relationship with the relationship 𝐴𝐵 (as defined in df-ss 2931). Other possible definitions are given by dfpss2 3029 and dfpss3 3030. (Contributed by NM, 7-Feb-1996.)
(𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
 
Theoremdfss2 2934* Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.)
(𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremdfss3 2935* Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
(𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
 
Theoremdfss2f 2936 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.)
𝑥𝐴    &   𝑥𝐵       (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremdfss3f 2937 Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)
𝑥𝐴    &   𝑥𝐵       (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
 
Theoremnfss 2938 If 𝑥 is not free in 𝐴 and 𝐵, it is not free in 𝐴𝐵. (Contributed by NM, 27-Dec-1996.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵
 
Theoremssel 2939 Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremssel2 2940 Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
((𝐴𝐵𝐶𝐴) → 𝐶𝐵)
 
Theoremsseli 2941 Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
𝐴𝐵       (𝐶𝐴𝐶𝐵)
 
Theoremsselii 2942 Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
𝐴𝐵    &   𝐶𝐴       𝐶𝐵
 
Theoremsseldi 2943 Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
𝐴𝐵    &   (𝜑𝐶𝐴)       (𝜑𝐶𝐵)
 
Theoremsseld 2944 Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
(𝜑𝐴𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))
 
Theoremsselda 2945 Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
(𝜑𝐴𝐵)       ((𝜑𝐶𝐴) → 𝐶𝐵)
 
Theoremsseldd 2946 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑𝐶𝐵)
 
Theoremssneld 2947 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.)
(𝜑𝐴𝐵)       (𝜑 → (¬ 𝐶𝐵 → ¬ 𝐶𝐴))
 
Theoremssneldd 2948 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.)
(𝜑𝐴𝐵)    &   (𝜑 → ¬ 𝐶𝐵)       (𝜑 → ¬ 𝐶𝐴)
 
Theoremssriv 2949* Inference rule based on subclass definition. (Contributed by NM, 5-Aug-1993.)
(𝑥𝐴𝑥𝐵)       𝐴𝐵
 
Theoremssrd 2950 Deduction rule based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴𝐵)
 
Theoremssrdv 2951* Deduction rule based on subclass definition. (Contributed by NM, 15-Nov-1995.)
(𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴𝐵)
 
Theoremsstr2 2952 Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(𝐴𝐵 → (𝐵𝐶𝐴𝐶))
 
Theoremsstr 2953 Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremsstri 2954 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
𝐴𝐵    &   𝐵𝐶       𝐴𝐶
 
Theoremsstrd 2955 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsyl5ss 2956 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
𝐴𝐵    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsyl6ss 2957 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)    &   𝐵𝐶       (𝜑𝐴𝐶)
 
Theoremsylan9ss 2958 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(𝜑𝐴𝐵)    &   (𝜓𝐵𝐶)       ((𝜑𝜓) → 𝐴𝐶)
 
Theoremsylan9ssr 2959 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
(𝜑𝐴𝐵)    &   (𝜓𝐵𝐶)       ((𝜓𝜑) → 𝐴𝐶)
 
Theoremeqss 2960 The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
 
Theoremeqssi 2961 Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)
𝐴𝐵    &   𝐵𝐴       𝐴 = 𝐵
 
Theoremeqssd 2962 Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐴)       (𝜑𝐴 = 𝐵)
 
Theoremeqrd 2963 Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremssid 2964 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.)
𝐴𝐴
 
Theoremssv 2965 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
𝐴 ⊆ V
 
Theoremsseq1 2966 Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremsseq2 2967 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremsseq12 2968 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))
 
Theoremsseq1i 2969 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)
 
Theoremsseq2i 2970 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)
 
Theoremsseq12i 2971 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       (𝐴𝐶𝐵𝐷)
 
Theoremsseq1d 2972 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))
 
Theoremsseq2d 2973 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))
 
Theoremsseq12d 2974 An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))
 
Theoremeqsstri 2975 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
𝐴 = 𝐵    &   𝐵𝐶       𝐴𝐶
 
Theoremeqsstr3i 2976 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
𝐵 = 𝐴    &   𝐵𝐶       𝐴𝐶
 
Theoremsseqtri 2977 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶
 
Theoremsseqtr4i 2978 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
𝐴𝐵    &   𝐶 = 𝐵       𝐴𝐶
 
Theoremeqsstrd 2979 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremeqsstr3d 2980 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐵 = 𝐴)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsseqtrd 2981 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)
 
Theoremsseqtr4d 2982 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)
 
Theorem3sstr3i 2983 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
𝐴𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐶𝐷
 
Theorem3sstr4i 2984 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
𝐴𝐵    &   𝐶 = 𝐴    &   𝐷 = 𝐵       𝐶𝐷
 
Theorem3sstr3g 2985 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
(𝜑𝐴𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝐷)
 
Theorem3sstr4g 2986 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝐷)
 
Theorem3sstr3d 2987 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝐷)
 
Theorem3sstr4d 2988 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝐷)
 
Theoremsyl5eqss 2989 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
𝐴 = 𝐵    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsyl5eqssr 2990 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
𝐵 = 𝐴    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)
 
Theoremsyl6sseq 2991 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴𝐵)    &   𝐵 = 𝐶       (𝜑𝐴𝐶)
 
Theoremsyl6sseqr 2992 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐵       (𝜑𝐴𝐶)
 
Theoremsyl5sseq 2993 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
𝐵𝐴    &   (𝜑𝐴 = 𝐶)       (𝜑𝐵𝐶)
 
Theoremsyl5sseqr 2994 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
𝐵𝐴    &   (𝜑𝐶 = 𝐴)       (𝜑𝐵𝐶)
 
Theoremsyl6eqss 2995 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   𝐵𝐶       (𝜑𝐴𝐶)
 
Theoremsyl6eqssr 2996 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝐵 = 𝐴)    &   𝐵𝐶       (𝜑𝐴𝐶)
 
Theoremeqimss 2997 Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(𝐴 = 𝐵𝐴𝐵)
 
Theoremeqimss2 2998 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)
(𝐵 = 𝐴𝐴𝐵)
 
Theoremeqimssi 2999 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
𝐴 = 𝐵       𝐴𝐵
 
Theoremeqimss2i 3000 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
𝐴 = 𝐵       𝐵𝐴
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