Nuprl Lemma : existse-between2_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-between2: ∃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
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
existse-between2: ∃e∈[e1,e2].P[e],
exists: ∃x:A. B[x],
member: t ∈ T,
cand: A c∧ B,
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
guard: {T}
Latex:
\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:
2016_05_16-AM-09_45_24
Last ObjectModification:
2015_12_28-PM-09_40_02
Theory : new!event-ordering
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