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Theorem List for Intuitionistic Logic Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsbcor 2801 Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)
([A / x](φ ψ) ↔ ([A / x]φ [A / x]ψ))

Theoremsbcorg 2802 Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.)
(A 𝑉 → ([A / x](φ ψ) ↔ ([A / x]φ [A / x]ψ)))

Theoremsbcbig 2803 Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)
(A 𝑉 → ([A / x](φψ) ↔ ([A / x]φ[A / x]ψ)))

Theoremsbcal 2804* Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)
([A / y]xφx[A / y]φ)

Theoremsbcalg 2805* Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
(A 𝑉 → ([A / y]xφx[A / y]φ))

Theoremsbcex2 2806* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
([A / y]xφx[A / y]φ)

Theoremsbcexg 2807* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
(A 𝑉 → ([A / y]xφx[A / y]φ))

Theoremsbceqal 2808* A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.)
(A 𝑉 → (x(x = Ax = B) → A = B))

Theoremsbeqalb 2809* Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)
(A 𝑉 → ((x(φx = A) x(φx = B)) → A = B))

Theoremsbcbid 2810 Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
xφ    &   (φ → (ψχ))       (φ → ([A / x]ψ[A / x]χ))

Theoremsbcbidv 2811* Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
(φ → (ψχ))       (φ → ([A / x]ψ[A / x]χ))

Theoremsbcbii 2812 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)
(φψ)       ([A / x]φ[A / x]ψ)

Theoremeqsbc3r 2813* eqsbc3 2796 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.)
(A B → ([A / x]𝐶 = x𝐶 = A))

Theoremsbc3ang 2814 Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(A 𝑉 → ([A / x](φ ψ χ) ↔ ([A / x]φ [A / x]ψ [A / x]χ)))

Theoremsbcel1gv 2815* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(A 𝑉 → ([A / x]x BA B))

Theoremsbcel2gv 2816* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(B 𝑉 → ([B / x]A xA B))

Theoremsbcimdv 2817* Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)
(φ → (ψχ))       ((φ A 𝑉) → ([A / x]ψ[A / x]χ))

Theoremsbctt 2818 Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)
((A 𝑉 xφ) → ([A / x]φφ))

Theoremsbcgf 2819 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
xφ       (A 𝑉 → ([A / x]φφ))

Theoremsbc19.21g 2820 Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)
xφ       (A 𝑉 → ([A / x](φψ) ↔ (φ[A / x]ψ)))

Theoremsbcg 2821* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 2819. (Contributed by Alan Sare, 10-Nov-2012.)
(A 𝑉 → ([A / x]φφ))

Theoremsbc2iegf 2822* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)
xψ    &   yψ    &   x B 𝑊    &   ((x = A y = B) → (φψ))       ((A 𝑉 B 𝑊) → ([A / x][B / y]φψ))

Theoremsbc2ie 2823* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)
A V    &   B V    &   ((x = A y = B) → (φψ))       ([A / x][B / y]φψ)

Theoremsbc2iedv 2824* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
A V    &   B V    &   (φ → ((x = A y = B) → (ψχ)))       (φ → ([A / x][B / y]ψχ))

Theoremsbc3ie 2825* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)
A V    &   B V    &   𝐶 V    &   ((x = A y = B z = 𝐶) → (φψ))       ([A / x][B / y][𝐶 / z]φψ)

Theoremsbccomlem 2826* Lemma for sbccom 2827. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
([A / x][B / y]φ[B / y][A / x]φ)

Theoremsbccom 2827* Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
([A / x][B / y]φ[B / y][A / x]φ)

Theoremsbcralt 2828* Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
((A 𝑉 yA) → ([A / x]y B φy B [A / x]φ))

Theoremsbcrext 2829* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
((A 𝑉 yA) → ([A / x]y B φy B [A / x]φ))

Theoremsbcralg 2830* Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(A 𝑉 → ([A / x]y B φy B [A / x]φ))

Theoremsbcrexg 2831* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(A 𝑉 → ([A / x]y B φy B [A / x]φ))

Theoremsbcreug 2832* Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.)
(A 𝑉 → ([A / x]∃!y B φ∃!y B [A / x]φ))

Theoremsbcabel 2833* Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
xB       (A 𝑉 → ([A / x]{yφ} B ↔ {y[A / x]φ} B))

Theoremrspsbc 2834* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This provides an axiom for a predicate calculus for a restricted domain. This theorem generalizes the unrestricted stdpc4 1655 and spsbc 2769. See also rspsbca 2835 and rspcsbela . (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
(A B → (x B φ[A / x]φ))

Theoremrspsbca 2835* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)
((A B x B φ) → [A / x]φ)

Theoremrspesbca 2836* Existence form of rspsbca 2835. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
((A B [A / x]φ) → x B φ)

Theoremspesbc 2837 Existence form of spsbc 2769. (Contributed by Mario Carneiro, 18-Nov-2016.)
([A / x]φxφ)

Theoremspesbcd 2838 form of spsbc 2769. (Contributed by Mario Carneiro, 9-Feb-2017.)
(φ[A / x]ψ)       (φxψ)

Theoremsbcth2 2839* A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
(x Bφ)       (A B[A / x]φ)

Theoremra5 2840 Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is an axiom of a predicate calculus for a restricted domain. Compare the unrestricted stdpc5 1473. (Contributed by NM, 16-Jan-2004.)
xφ       (x A (φψ) → (φx A ψ))

Theoremrmo2ilem 2841* Condition implying restricted "at most one." (Contributed by Jim Kingdon, 14-Jul-2018.)
yφ       (yx A (φx = y) → ∃*x A φ)

Theoremrmo2i 2842* Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)
yφ       (y A x A (φx = y) → ∃*x A φ)

Theoremrmo3 2843* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
yφ       (∃*x A φx A y A ((φ [y / x]φ) → x = y))

Theoremrmob 2844* Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)
(x = B → (φψ))    &   (x = 𝐶 → (φχ))       ((∃*x A φ (B A ψ)) → (B = 𝐶 ↔ (𝐶 A χ)))

Theoremrmoi 2845* Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
(x = B → (φψ))    &   (x = 𝐶 → (φχ))       ((∃*x A φ (B A ψ) (𝐶 A χ)) → B = 𝐶)

2.1.10  Proper substitution of classes for sets into classes

Syntaxcsb 2846 Extend class notation to include the proper substitution of a class for a set into another class.
class A / xB

Definitiondf-csb 2847* Define the proper substitution of a class for a set into another class. The underlined brackets distinguish it from the substitution into a wff, wsbc 2758, to prevent ambiguity. Theorem sbcel1g 2863 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 2872 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
A / xB = {y[A / x]y B}

Theoremcsb2 2848* Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that x can be free in B but cannot occur in A. (Contributed by NM, 2-Dec-2013.)
A / xB = {yx(x = A y B)}

Theoremcsbeq1 2849 Analog of dfsbcq 2760 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
(A = BA / x𝐶 = B / x𝐶)

Theoremcbvcsb 2850 Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on A. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.)
y𝐶    &   x𝐷    &   (x = y𝐶 = 𝐷)       A / x𝐶 = A / y𝐷

Theoremcbvcsbv 2851* Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
(x = yB = 𝐶)       A / xB = A / y𝐶

Theoremcsbeq1d 2852 Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)
(φA = B)       (φA / x𝐶 = B / x𝐶)

Theoremcsbid 2853 Analog of sbid 1654 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
x / xA = A

Theoremcsbeq1a 2854 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
(x = AB = A / xB)

Theoremcsbco 2855* Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
A / yy / xB = A / xB

Theoremcsbtt 2856 Substitution doesn't affect a constant B (in which x is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)
((A 𝑉 xB) → A / xB = B)

Theoremcsbconstgf 2857 Substitution doesn't affect a constant B (in which x is not free). (Contributed by NM, 10-Nov-2005.)
xB       (A 𝑉A / xB = B)

Theoremcsbconstg 2858* Substitution doesn't affect a constant B (in which x is not free). csbconstgf 2857 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)
(A 𝑉A / xB = B)

Theoremsbcel12g 2859 Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(A 𝑉 → ([A / x]B 𝐶A / xB A / x𝐶))

Theoremsbceqg 2860 Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(A 𝑉 → ([A / x]B = 𝐶A / xB = A / x𝐶))

Theoremsbcnel12g 2861 Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
(A 𝑉 → ([A / x]B𝐶A / xBA / x𝐶))

Theoremsbcne12g 2862 Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)
(A 𝑉 → ([A / x]B𝐶A / xBA / x𝐶))

Theoremsbcel1g 2863* Move proper substitution in and out of a membership relation. Note that the scope of [A / x] is the wff B 𝐶, whereas the scope of A / x is the class B. (Contributed by NM, 10-Nov-2005.)
(A 𝑉 → ([A / x]B 𝐶A / xB 𝐶))

Theoremsbceq1g 2864* Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)
(A 𝑉 → ([A / x]B = 𝐶A / xB = 𝐶))

Theoremsbcel2g 2865* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)
(A 𝑉 → ([A / x]B 𝐶B A / x𝐶))

Theoremsbceq2g 2866* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)
(A 𝑉 → ([A / x]B = 𝐶B = A / x𝐶))

Theoremcsbcomg 2867* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
((A 𝑉 B 𝑊) → A / xB / y𝐶 = B / yA / x𝐶)

Theoremcsbeq2d 2868 Formula-building deduction rule for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
xφ    &   (φB = 𝐶)       (φA / xB = A / x𝐶)

Theoremcsbeq2dv 2869* Formula-building deduction rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
(φB = 𝐶)       (φA / xB = A / x𝐶)

Theoremcsbeq2i 2870 Formula-building inference rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
B = 𝐶       A / xB = A / x𝐶

Theoremcsbvarg 2871 The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)
(A 𝑉A / xx = A)

Theoremsbccsbg 2872* Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)
(A 𝑉 → ([A / x]φy A / x{yφ}))

Theoremsbccsb2g 2873 Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)
(A 𝑉 → ([A / x]φA A / x{xφ}))

Theoremnfcsb1d 2874 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
(φxA)       (φxA / xB)

Theoremnfcsb1 2875 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
xA       xA / xB

Theoremnfcsb1v 2876* Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
xA / xB

Theoremnfcsbd 2877 Deduction version of nfcsb 2878. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
yφ    &   (φxA)    &   (φxB)       (φxA / yB)

Theoremnfcsb 2878 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
xA    &   xB       xA / yB

Theoremcsbhypf 2879* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2597 for class substitution version. (Contributed by NM, 19-Dec-2008.)
xA    &   x𝐶    &   (x = AB = 𝐶)       (y = Ay / xB = 𝐶)

Theoremcsbiebt 2880* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2884.) (Contributed by NM, 11-Nov-2005.)
((A 𝑉 x𝐶) → (x(x = AB = 𝐶) ↔ A / xB = 𝐶))

Theoremcsbiedf 2881* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)
xφ    &   (φx𝐶)    &   (φA 𝑉)    &   ((φ x = A) → B = 𝐶)       (φA / xB = 𝐶)

Theoremcsbieb 2882* Bidirectional conversion between an implicit class substitution hypothesis x = AB = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)
A V    &   x𝐶       (x(x = AB = 𝐶) ↔ A / xB = 𝐶)

Theoremcsbiebg 2883* Bidirectional conversion between an implicit class substitution hypothesis x = AB = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
x𝐶       (A 𝑉 → (x(x = AB = 𝐶) ↔ A / xB = 𝐶))

Theoremcsbiegf 2884* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
(A 𝑉x𝐶)    &   (x = AB = 𝐶)       (A 𝑉A / xB = 𝐶)

Theoremcsbief 2885* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
A V    &   x𝐶    &   (x = AB = 𝐶)       A / xB = 𝐶

Theoremcsbied 2886* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)
(φA 𝑉)    &   ((φ x = A) → B = 𝐶)       (φA / xB = 𝐶)

Theoremcsbied2 2887* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
(φA 𝑉)    &   (φA = B)    &   ((φ x = B) → 𝐶 = 𝐷)       (φA / x𝐶 = 𝐷)

Theoremcsbie2t 2888* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 2889). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)
A V    &   B V       (xy((x = A y = B) → 𝐶 = 𝐷) → A / xB / y𝐶 = 𝐷)

Theoremcsbie2 2889* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)
A V    &   B V    &   ((x = A y = B) → 𝐶 = 𝐷)       A / xB / y𝐶 = 𝐷

Theoremcsbie2g 2890* Conversion of implicit substitution to explicit class substitution. This version of sbcie 2791 avoids a disjointness condition on x and A by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
(x = yB = 𝐶)    &   (y = A𝐶 = 𝐷)       (A 𝑉A / xB = 𝐷)

Theoremsbcnestgf 2891 Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
((A 𝑉 yxφ) → ([A / x][B / y]φ[A / xB / y]φ))

Theoremcsbnestgf 2892 Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
((A 𝑉 yx𝐶) → A / xB / y𝐶 = A / xB / y𝐶)

Theoremsbcnestg 2893* Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
(A 𝑉 → ([A / x][B / y]φ[A / xB / y]φ))

Theoremcsbnestg 2894* Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
(A 𝑉A / xB / y𝐶 = A / xB / y𝐶)

Theoremcsbnest1g 2895 Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
(A 𝑉A / xB / x𝐶 = A / xB / x𝐶)

Theoremcsbidmg 2896* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)
(A 𝑉A / xA / xB = A / xB)

Theoremsbcco3g 2897* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
(x = AB = 𝐶)       (A 𝑉 → ([A / x][B / y]φ[𝐶 / y]φ))

Theoremcsbco3g 2898* Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
(x = AB = 𝐶)       (A 𝑉A / xB / y𝐷 = 𝐶 / y𝐷)

Theoremrspcsbela 2899* Special case related to rspsbc 2834. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
((A B x B 𝐶 𝐷) → A / x𝐶 𝐷)

Theoremsbnfc2 2900* Two ways of expressing "x is (effectively) not free in A." (Contributed by Mario Carneiro, 14-Oct-2016.)
(xAyzy / xA = z / xA)

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