Nuprl Lemma : FormSafe1-Fvs-subset
∀C:Type. ∀phi:Form(C). ∀vs:Atom List.  ((FormSafe1(phi) vs) 
⇒ l_subset(Atom;vs;FormFvs(phi)))
Proof
Definitions occuring in Statement : 
FormSafe1: FormSafe1(f)
, 
FormFvs: FormFvs(f)
, 
Form: Form(C)
, 
l_subset: l_subset(T;as;bs)
, 
list: T List
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
apply: f a
, 
atom: Atom
, 
universe: Type
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
so_lambda: λ2x.t[x]
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
so_apply: x[s]
, 
FormFvs: FormFvs(f)
, 
FormSafe1: FormSafe1(f)
, 
FormVar: Vname
, 
Form_ind: Form_ind, 
false: False
, 
FormConst: Const(value)
, 
FormSet: {var | phi}
, 
FormEqual: left = right
, 
subtype_rel: A ⊆r B
, 
uimplies: b supposing a
, 
top: Top
, 
and: P ∧ Q
, 
or: P ∨ Q
, 
FormMember: element ∈ set
, 
FormAnd: left ∧ right)
, 
FormOr: left ∨ right
, 
FormNot: ¬(body)
, 
FormAll: ∀var. body
, 
FormExists: ∃var. body
, 
guard: {T}
, 
uiff: uiff(P;Q)
, 
squash: ↓T
, 
true: True
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
exists: ∃x:A. B[x]
, 
l_subset: l_subset(T;as;bs)
, 
sq_type: SQType(T)
, 
set-equal: set-equal(T;x;y)
, 
ext-eq: A ≡ B
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
eq_atom: x =a y
, 
ifthenelse: if b then t else f fi 
, 
FormVar-name: FormVar-name(v)
, 
pi2: snd(t)
, 
FormVar?: FormVar?(v)
, 
pi1: fst(t)
, 
assert: ↑b
, 
bfalse: ff
, 
bnot: ¬bb
, 
not: ¬A
Lemmas referenced : 
Form-induction, 
all_wf, 
list_wf, 
FormSafe1_wf, 
l_subset_wf, 
FormFvs_wf, 
Form_wf, 
false_wf, 
or_wf, 
assert_wf, 
null_wf3, 
subtype_rel_list, 
top_wf, 
exists_wf, 
set-equal_wf, 
cons_wf, 
nil_wf, 
FormVar?_wf, 
equal-wf-T-base, 
FormVar-name_wf, 
atom_subtype_base, 
not_wf, 
l_member_wf, 
append_wf, 
l_disjoint_wf, 
assert_of_null, 
squash_wf, 
true_wf, 
l-union_wf, 
atom-deq_wf, 
iff_weakening_equal, 
l_subset_nil_left, 
subtype_base_sq, 
member_singleton, 
member-union, 
Form-ext, 
eq_atom_wf, 
bool_wf, 
eqtt_to_assert, 
assert_of_eq_atom, 
cons_member, 
eqff_to_assert, 
equal_wf, 
bool_cases_sqequal, 
bool_subtype_base, 
assert-bnot, 
neg_assert_of_eq_atom, 
member_append, 
equal-wf-base, 
member_filter, 
iff_transitivity, 
bnot_wf, 
iff_weakening_uiff, 
assert_of_bnot, 
and_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
hypothesis, 
sqequalRule, 
lambdaEquality, 
atomEquality, 
functionEquality, 
applyEquality, 
cumulativity, 
independent_functionElimination, 
voidElimination, 
because_Cache, 
independent_isectElimination, 
isect_memberEquality, 
voidEquality, 
productEquality, 
productElimination, 
universeEquality, 
unionElimination, 
imageElimination, 
equalityTransitivity, 
equalitySymmetry, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
instantiate, 
dependent_functionElimination, 
inlFormation, 
inrFormation, 
promote_hyp, 
hypothesis_subsumption, 
tokenEquality, 
equalityElimination, 
dependent_pairFormation, 
independent_pairFormation, 
impliesFunctionality, 
hyp_replacement, 
dependent_set_memberEquality, 
applyLambdaEquality, 
setElimination, 
rename
Latex:
\mforall{}C:Type.  \mforall{}phi:Form(C).  \mforall{}vs:Atom  List.    ((FormSafe1(phi)  vs)  {}\mRightarrow{}  l\_subset(Atom;vs;FormFvs(phi)))
Date html generated:
2018_05_21-PM-11_28_44
Last ObjectModification:
2017_10_12-AM-00_45_16
Theory : PZF
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