### Nuprl Lemma : l_exists_iff

`∀[T:Type]. ∀L:T List. ∀[P:{x:T| (x ∈ L)}  ⟶ ℙ]. ((∃x∈L. P[x]) `⇐⇒` ∃x:T. ((x ∈ L) ∧ P[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` so_apply: `x[s]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  l_exists: `(∃x∈L. P[x])` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` int_seg: `{i..j-}` sq_stable: `SqStable(P)` lelt: `i ≤ j < k` squash: `↓T` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` exists: `∃x:A. B[x]` cand: `A c∧ B` l_member: `(x ∈ l)` nat: `ℕ` le: `A ≤ B` label: `...\$L... t` guard: `{T}`
Lemmas referenced :  exists_wf int_seg_wf length_wf l_member_wf select_wf list-subtype sq_stable__le list_wf select_member lelt_wf less_than_wf equal_wf iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation independent_pairFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality cumulativity hypothesisEquality hypothesis lambdaEquality applyEquality functionExtensionality setEquality because_Cache equalityTransitivity equalitySymmetry independent_isectElimination setElimination rename independent_functionElimination productElimination imageMemberEquality baseClosed imageElimination productEquality dependent_set_memberEquality universeEquality functionEquality dependent_pairFormation dependent_functionElimination

Latex:
\mforall{}[T:Type].  \mforall{}L:T  List.  \mforall{}[P:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbP{}].  ((\mexists{}x\mmember{}L.  P[x])  \mLeftarrow{}{}\mRightarrow{}  \mexists{}x:T.  ((x  \mmember{}  L)  \mwedge{}  P[x]))

Date html generated: 2017_04_14-AM-08_40_11
Last ObjectModification: 2017_02_27-PM-03_30_49

Theory : list_0

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