Nuprl Lemma : classical-exists2
∀[T:Type]. ∀[P:T ⟶ ℙ].  uiff(¬(∀x:T. (¬P[x]));{∃x:T. P[x]})
Proof
Definitions occuring in Statement : 
classical: {P}
, 
uiff: uiff(P;Q)
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
not: ¬A
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
uimplies: b supposing a
, 
classical: {P}
, 
unit: Unit
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
exists: ∃x:A. B[x]
, 
prop: ℙ
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
false: False
, 
all: ∀x:A. B[x]
, 
or: P ∨ Q
Lemmas referenced : 
it_wf, 
classical-excluded-middle, 
classical_wf, 
all_wf, 
not_wf, 
exists_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
independent_pairFormation, 
sqequalRule, 
sqequalHypSubstitution, 
setElimination, 
thin, 
rename, 
dependent_set_memberEquality, 
axiomEquality, 
natural_numberEquality, 
hypothesis, 
lemma_by_obid, 
isectElimination, 
hypothesisEquality, 
lambdaEquality, 
applyEquality, 
lambdaFormation, 
independent_functionElimination, 
voidElimination, 
dependent_functionElimination, 
productElimination, 
independent_pairEquality, 
isect_memberEquality, 
because_Cache, 
equalityTransitivity, 
equalitySymmetry, 
functionEquality, 
cumulativity, 
universeEquality, 
unionElimination, 
dependent_pairFormation
Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbP{}].    uiff(\mneg{}(\mforall{}x:T.  (\mneg{}P[x]));\{\mexists{}x:T.  P[x]\})
Date html generated:
2016_05_13-PM-03_17_01
Last ObjectModification:
2016_01_06-PM-05_20_32
Theory : core_2
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