Nuprl Lemma : l_exists_functionality
∀[T:Type]
∀L:T List. ∀[P,Q:{x:T| (x ∈ L)} ⟶ ℙ]. ((∀x:{x:T| (x ∈ L)} . (P[x] ⇐⇒ Q[x])) ⇒ {(∃x∈L. P[x]) ⇐⇒ (∃x∈L. Q[x])})
Proof
Definitions occuring in Statement :
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
uimplies: b supposing a,
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T,
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q
Lemmas referenced :
list_wf,
iff_wf,
all_wf,
l_exists_wf,
sq_stable__le,
list-subtype,
l_member_wf,
select_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
cut,
hypothesis,
dependent_functionElimination,
lemma_by_obid,
isectElimination,
setEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
setElimination,
rename,
natural_numberEquality,
independent_functionElimination,
introduction,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
lambdaEquality,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}L:T List
\mforall{}[P,Q:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}].
((\mforall{}x:\{x:T| (x \mmember{} L)\} . (P[x] \mLeftarrow{}{}\mRightarrow{} Q[x])) {}\mRightarrow{} \{(\mexists{}x\mmember{}L. P[x]) \mLeftarrow{}{}\mRightarrow{} (\mexists{}x\mmember{}L. Q[x])\})
Date html generated:
2016_05_14-AM-06_40_13
Last ObjectModification:
2016_01_14-PM-08_20_50
Theory : list_0
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