### Nuprl Lemma : l_all_single

`∀[T:Type]. ∀t:T. ∀[P:{x:T| x = t ∈ T}  ⟶ ℙ]. ((∀x∈[t].P[x]) `⇐⇒` P[t])`

Proof

Definitions occuring in Statement :  l_all: `(∀x∈L.P[x])` cons: `[a / b]` nil: `[]` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  l_all: `(∀x∈L.P[x])` all: `∀x:A. B[x]` member: `t ∈ T` top: `Top` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` 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` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` false: `False` uiff: `uiff(P;Q)` subtract: `n - m` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` less_than: `a < b`
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin isect_memberEquality voidElimination voidEquality hypothesis isect_memberFormation lambdaFormation independent_pairFormation isectElimination natural_numberEquality lambdaEquality applyEquality functionExtensionality hypothesisEquality setEquality because_Cache dependent_set_memberEquality cumulativity independent_isectElimination setElimination rename independent_functionElimination productElimination imageMemberEquality baseClosed imageElimination unionElimination addEquality minusEquality intEquality functionEquality universeEquality

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

Date html generated: 2017_04_14-AM-08_40_06
Last ObjectModification: 2017_02_27-PM-03_30_52

Theory : list_0

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