### Nuprl Lemma : existse-between1_functionality_wrt_iff

`∀es:EO. ∀e1:E. ∀e2:{e:E| loc(e) = loc(e1) ∈ Id} .`
`  ∀[P,Q:{e:E| loc(e) = loc(e1) ∈ Id}  ─→ ℙ].`
`    ((∀e:{e:E| loc(e) = loc(e1) ∈ Id} . (P[e] `⇐⇒` Q[e])) `` (∃e∈[e1,e2).P[e] `⇐⇒` ∃e∈[e1,e2).Q[e]))`

Proof

Definitions occuring in Statement :  existse-between1: `∃e∈[e1,e2).P[e]` es-loc: `loc(e)` es-E: `E` event_ordering: `EO` Id: `Id` 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]` equal: `s = t ∈ T`
Lemmas :  es-le_wf es-locl_wf Id_wf es-loc_wf existse-between1_wf all_wf iff_wf es-E_wf set_wf event_ordering_wf es-le-loc
\mforall{}es:EO.  \mforall{}e1:E.  \mforall{}e2:\{e:E|  loc(e)  =  loc(e1)\}  .
\mforall{}[P,Q:\{e:E|  loc(e)  =  loc(e1)\}    {}\mrightarrow{}  \mBbbP{}].
((\mforall{}e:\{e:E|  loc(e)  =  loc(e1)\}  .  (P[e]  \mLeftarrow{}{}\mRightarrow{}  Q[e]))  {}\mRightarrow{}  (\mexists{}e\mmember{}[e1,e2).P[e]  \mLeftarrow{}{}\mRightarrow{}  \mexists{}e\mmember{}[e1,e2).Q[e]))

Date html generated: 2015_07_17-AM-08_47_30
Last ObjectModification: 2015_01_27-PM-02_26_41

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