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Theorem List for New Foundations Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrmobiia 2801 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
(x A → (φψ))       (∃*x Aφ∃*x Aψ)
 
Theoremrmobii 2802 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
(φψ)       (∃*x Aφ∃*x Aψ)
 
Theoremraleqf 2803 Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
F/_xA    &   F/_xB       (A = B → (x A φx B φ))
 
Theoremrexeqf 2804 Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)
F/_xA    &   F/_xB       (A = B → (x A φx B φ))
 
Theoremreueq1f 2805 Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
F/_xA    &   F/_xB       (A = B → (∃!x A φ∃!x B φ))
 
Theoremrmoeq1f 2806 Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
F/_xA    &   F/_xB       (A = B → (∃*x Aφ∃*x Bφ))
 
Theoremraleq 2807* Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
(A = B → (x A φx B φ))
 
Theoremrexeq 2808* Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
(A = B → (x A φx B φ))
 
Theoremreueq1 2809* Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
(A = B → (∃!x A φ∃!x B φ))
 
Theoremrmoeq1 2810* Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(A = B → (∃*x Aφ∃*x Bφ))
 
Theoremraleqi 2811* Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
A = B       (x A φx B φ)
 
Theoremrexeqi 2812* Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
A = B       (x A φx B φ)
 
Theoremraleqdv 2813* Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
(φA = B)       (φ → (x A ψx B ψ))
 
Theoremrexeqdv 2814* Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
(φA = B)       (φ → (x A ψx B ψ))
 
Theoremraleqbi1dv 2815* Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
(A = B → (φψ))       (A = B → (x A φx B ψ))
 
Theoremrexeqbi1dv 2816* Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
(A = B → (φψ))       (A = B → (x A φx B ψ))
 
Theoremreueqd 2817* Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
(A = B → (φψ))       (A = B → (∃!x A φ∃!x B ψ))
 
Theoremrmoeqd 2818* Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(A = B → (φψ))       (A = B → (∃*x Aφ∃*x Bψ))
 
Theoremraleqbidv 2819* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
(φA = B)    &   (φ → (ψχ))       (φ → (x A ψx B χ))
 
Theoremrexeqbidv 2820* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
(φA = B)    &   (φ → (ψχ))       (φ → (x A ψx B χ))
 
Theoremraleqbidva 2821* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
(φA = B)    &   ((φ x A) → (ψχ))       (φ → (x A ψx B χ))
 
Theoremrexeqbidva 2822* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
(φA = B)    &   ((φ x A) → (ψχ))       (φ → (x A ψx B χ))
 
Theoremmormo 2823 Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
(∃*xφ∃*x Aφ)
 
Theoremreu5 2824 Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)
(∃!x A φ ↔ (x A φ ∃*x Aφ))
 
Theoremreurex 2825 Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
(∃!x A φx A φ)
 
Theoremreurmo 2826 Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
(∃!x A φ∃*x Aφ)
 
Theoremrmo5 2827 Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
(∃*x Aφ ↔ (x A φ∃!x A φ))
 
Theoremnrexrmo 2828 Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
x A φ∃*x Aφ)
 
Theoremcbvralf 2829 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
F/_xA    &   F/_yA    &   yφ    &   xψ    &   (x = y → (φψ))       (x A φy A ψ)
 
Theoremcbvrexf 2830 Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.)
F/_xA    &   F/_yA    &   yφ    &   xψ    &   (x = y → (φψ))       (x A φy A ψ)
 
Theoremcbvral 2831* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)
yφ    &   xψ    &   (x = y → (φψ))       (x A φy A ψ)
 
Theoremcbvrex 2832* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
yφ    &   xψ    &   (x = y → (φψ))       (x A φy A ψ)
 
Theoremcbvreu 2833* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)
yφ    &   xψ    &   (x = y → (φψ))       (∃!x A φ∃!y A ψ)
 
Theoremcbvrmo 2834* Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)
yφ    &   xψ    &   (x = y → (φψ))       (∃*x Aφ∃*y Aψ)
 
Theoremcbvralv 2835* Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.)
(x = y → (φψ))       (x A φy A ψ)
 
Theoremcbvrexv 2836* Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.)
(x = y → (φψ))       (x A φy A ψ)
 
Theoremcbvreuv 2837* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
(x = y → (φψ))       (∃!x A φ∃!y A ψ)
 
Theoremcbvrmov 2838* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(x = y → (φψ))       (∃*x Aφ∃*y Aψ)
 
Theoremcbvraldva2 2839* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
((φ x = y) → (ψχ))    &   ((φ x = y) → A = B)       (φ → (x A ψy B χ))
 
Theoremcbvrexdva2 2840* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
((φ x = y) → (ψχ))    &   ((φ x = y) → A = B)       (φ → (x A ψy B χ))
 
Theoremcbvraldva 2841* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
((φ x = y) → (ψχ))       (φ → (x A ψy A χ))
 
Theoremcbvrexdva 2842* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
((φ x = y) → (ψχ))       (φ → (x A ψy A χ))
 
Theoremcbvral2v 2843* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)
(x = z → (φχ))    &   (y = w → (χψ))       (x A y B φz A w B ψ)
 
Theoremcbvrex2v 2844* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)
(x = z → (φχ))    &   (y = w → (χψ))       (x A y B φz A w B ψ)
 
Theoremcbvral3v 2845* Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)
(x = w → (φχ))    &   (y = v → (χθ))    &   (z = u → (θψ))       (x A y B z C φw A v B u C ψ)
 
Theoremcbvralsv 2846* Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)
(x A φy A [y / x]φ)
 
Theoremcbvrexsv 2847* Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)
(x A φy A [y / x]φ)
 
Theoremsbralie 2848* Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)
(x = y → (φψ))       ([x / y]x y φy x ψ)
 
Theoremrabbiia 2849 Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)
(x A → (φψ))       {x A φ} = {x A ψ}
 
Theoremrabbidva 2850* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)
((φ x A) → (ψχ))       (φ → {x A ψ} = {x A χ})
 
Theoremrabbidv 2851* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.)
(φ → (ψχ))       (φ → {x A ψ} = {x A χ})
 
Theoremrabeqf 2852 Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)
F/_xA    &   F/_xB       (A = B → {x A φ} = {x B φ})
 
Theoremrabeq 2853* Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
(A = B → {x A φ} = {x B φ})
 
Theoremrabeqbidv 2854* Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
(φA = B)    &   (φ → (ψχ))       (φ → {x A ψ} = {x B χ})
 
Theoremrabeqbidva 2855* Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
(φA = B)    &   ((φ x A) → (ψχ))       (φ → {x A ψ} = {x B χ})
 
Theoremrabeq2i 2856 Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
A = {x B φ}       (x A ↔ (x B φ))
 
Theoremcbvrab 2857 Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.)
F/_xA    &   F/_yA    &   yφ    &   xψ    &   (x = y → (φψ))       {x A φ} = {y A ψ}
 
Theoremcbvrabv 2858* Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.)
(x = y → (φψ))       {x A φ} = {y A ψ}
 
2.1.6  The universal class
 
Syntaxcvv 2859 Extend class notation to include the universal class symbol.
class V
 
Theoremvjust 2860 Soundness justification theorem for df-v 2861. (Contributed by Rodolfo Medina, 27-Apr-2010.)
{x x = x} = {y y = y}
 
Definitiondf-v 2861 Define the universal class. Definition 5.20 of [TakeutiZaring] p. 21. Also Definition 2.9 of [Quine] p. 19. (Contributed by NM, 5-Aug-1993.)
V = {x x = x}
 
Theoremvex 2862 All set variables are sets (see isset 2863). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.)
x V
 
Theoremisset 2863* Two ways to say "A is a set": A class A is a member of the universal class V (see df-v 2861) if and only if the class A exists (i.e. there exists some set x equal to class A). Theorem 6.9 of [Quine] p. 43. Notational convention: We will use the notational device "A V " to mean "A is a set" very frequently, for example in uniex 4317. Note the when A is not a set, it is called a proper class. In some theorems, such as uniexg 4316, in order to shorten certain proofs we use the more general antecedent A V instead of A V to mean "A is a set."

Note that a constant is implicitly considered distinct from all variables. This is why V is not included in the distinct variable list, even though df-clel 2349 requires that the expression substituted for B not contain x. (Also, the Metamath spec does not allow constants in the distinct variable list.) (Contributed by NM, 5-Aug-1993.)

(A V ↔ x x = A)
 
Theoremissetf 2864 A version of isset that does not require x and A to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 10-Oct-2016.)
F/_xA       (A V ↔ x x = A)
 
Theoremisseti 2865* A way to say "A is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
A V       x x = A
 
Theoremissetri 2866* A way to say "A is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
x x = A       A V
 
Theoremelex 2867 If a class is a member of another class, it is a set. Theorem 6.12 of [Quine] p. 44. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(A BA V)
 
Theoremelexi 2868 If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)
A B       A V
 
Theoremelisset 2869* An element of a class exists. (Contributed by NM, 1-May-1995.)
(A Vx x = A)
 
Theoremelex22 2870* If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
((A B A C) → x(x B x C))
 
Theoremelex2 2871* If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)
(A Bx x B)
 
Theoremralv 2872 A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
(x V φxφ)
 
Theoremrexv 2873 An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
(x V φxφ)
 
Theoremreuv 2874 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
(∃!x V φ∃!xφ)
 
Theoremrmov 2875 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∃*x Vφ∃*xφ)
 
Theoremrabab 2876 A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
{x V φ} = {x φ}
 
Theoremralcom4 2877* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(x A yφyx A φ)
 
Theoremrexcom4 2878* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(x A yφyx A φ)
 
Theoremrexcom4a 2879* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
(xy A (φ ψ) ↔ y A (φ xψ))
 
Theoremrexcom4b 2880* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
B V       (xy A (φ x = B) ↔ y A φ)
 
Theoremceqsalt 2881* Closed theorem version of ceqsalg 2883. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
((Ⅎxψ x(x = A → (φψ)) A V) → (x(x = Aφ) ↔ ψ))
 
Theoremceqsralt 2882* Restricted quantifier version of ceqsalt 2881. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
((Ⅎxψ x(x = A → (φψ)) A B) → (x B (x = Aφ) ↔ ψ))
 
Theoremceqsalg 2883* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
xψ    &   (x = A → (φψ))       (A V → (x(x = Aφ) ↔ ψ))
 
Theoremceqsal 2884* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
xψ    &   A V    &   (x = A → (φψ))       (x(x = Aφ) ↔ ψ)
 
Theoremceqsalv 2885* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
A V    &   (x = A → (φψ))       (x(x = Aφ) ↔ ψ)
 
Theoremceqsralv 2886* Restricted quantifier version of ceqsalv 2885. (Contributed by NM, 21-Jun-2013.)
(x = A → (φψ))       (A B → (x B (x = Aφ) ↔ ψ))
 
Theoremgencl 2887* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(θx(χ A = B))    &   (A = B → (φψ))    &   (χφ)       (θψ)
 
Theorem2gencl 2888* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(C Sx R A = C)    &   (D Sy R B = D)    &   (A = C → (φψ))    &   (B = D → (ψχ))    &   ((x R y R) → φ)       ((C S D S) → χ)
 
Theorem3gencl 2889* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(D Sx R A = D)    &   (F Sy R B = F)    &   (G Sz R C = G)    &   (A = D → (φψ))    &   (B = F → (ψχ))    &   (C = G → (χθ))    &   ((x R y R z R) → φ)       ((D S F S G S) → θ)
 
Theoremcgsexg 2890* Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
(x = Aχ)    &   (χ → (φψ))       (A V → (x(χ φ) ↔ ψ))
 
Theoremcgsex2g 2891* Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)
((x = A y = B) → χ)    &   (χ → (φψ))       ((A V B W) → (xy(χ φ) ↔ ψ))
 
Theoremcgsex4g 2892* An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)
(((x = A y = B) (z = C w = D)) → χ)    &   (χ → (φψ))       (((A R B S) (C R D S)) → (xyzw(χ φ) ↔ ψ))
 
Theoremceqsex 2893* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)
xψ    &   A V    &   (x = A → (φψ))       (x(x = A φ) ↔ ψ)
 
Theoremceqsexv 2894* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)
A V    &   (x = A → (φψ))       (x(x = A φ) ↔ ψ)
 
Theoremceqsex2 2895* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
xψ    &   yχ    &   A V    &   B V    &   (x = A → (φψ))    &   (y = B → (ψχ))       (xy(x = A y = B φ) ↔ χ)
 
Theoremceqsex2v 2896* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
A V    &   B V    &   (x = A → (φψ))    &   (y = B → (ψχ))       (xy(x = A y = B φ) ↔ χ)
 
Theoremceqsex3v 2897* Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)
A V    &   B V    &   C V    &   (x = A → (φψ))    &   (y = B → (ψχ))    &   (z = C → (χθ))       (xyz((x = A y = B z = C) φ) ↔ θ)
 
Theoremceqsex4v 2898* Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
A V    &   B V    &   C V    &   D V    &   (x = A → (φψ))    &   (y = B → (ψχ))    &   (z = C → (χθ))    &   (w = D → (θτ))       (xyzw((x = A y = B) (z = C w = D) φ) ↔ τ)
 
Theoremceqsex6v 2899* Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)
A V    &   B V    &   C V    &   D V    &   E V    &   F V    &   (x = A → (φψ))    &   (y = B → (ψχ))    &   (z = C → (χθ))    &   (w = D → (θτ))    &   (v = E → (τη))    &   (u = F → (ηζ))       (xyzwvu((x = A y = B z = C) (w = D v = E u = F) φ) ↔ ζ)
 
Theoremceqsex8v 2900* Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
A V    &   B V    &   C V    &   D V    &   E V    &   F V    &   G V    &   H V    &   (x = A → (φψ))    &   (y = B → (ψχ))    &   (z = C → (χθ))    &   (w = D → (θτ))    &   (v = E → (τη))    &   (u = F → (ηζ))    &   (t = G → (ζσ))    &   (s = H → (σρ))       (xyzwvuts(((x = A y = B) (z = C w = D)) ((v = E u = F) (t = G s = H)) φ) ↔ ρ)
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