### 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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