### Nuprl Lemma : l_all_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_all: `(∀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]` 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]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` l_member: `(x ∈ l)` exists: `∃x:A. B[x]` l_all: `(∀x∈L.P[x])` nat: `ℕ` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` false: `False` uiff: `uiff(P;Q)` uimplies: `b supposing a` cand: `A c∧ B` subtype_rel: `A ⊆r B` top: `Top` less_than': `less_than'(a;b)` true: `True` sq_stable: `SqStable(P)` squash: `↓T`
Lemmas referenced :  l_member_wf l_all_wf all_wf list_wf decidable__lt length_wf false_wf not-lt-2 less-iff-le add_functionality_wrt_le add-swap add-commutes le-add-cancel lelt_wf select_wf list-subtype sq_stable__le nat_wf select_member int_seg_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation independent_pairFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality hypothesis sqequalRule lambdaEquality applyEquality functionExtensionality setEquality setElimination rename dependent_set_memberEquality functionEquality because_Cache universeEquality productElimination dependent_functionElimination unionElimination voidElimination independent_functionElimination independent_isectElimination addEquality natural_numberEquality isect_memberEquality voidEquality intEquality addLevel levelHypothesis equalityTransitivity equalitySymmetry imageMemberEquality baseClosed imageElimination hyp_replacement Error :applyLambdaEquality

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

Date html generated: 2016_10_21-AM-09_48_59
Last ObjectModification: 2016_07_12-AM-05_08_38

Theory : list_0

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