Nuprl Lemma : exists_over_and_r
∀[T:Type]. ∀[A:ℙ]. ∀[B:T ⟶ ℙ]. (∃x:T. (A ∧ B[x]) ⇐⇒ A ∧ (∃x:T. B[x]))
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q
Lemmas referenced :
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
independent_pairFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
dependent_pairFormation,
hypothesisEquality,
applyEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
productEquality,
cumulativity,
universeEquality,
because_Cache,
Error :functionIsType,
Error :universeIsType,
Error :inhabitedIsType
Latex:
\mforall{}[T:Type]. \mforall{}[A:\mBbbP{}]. \mforall{}[B:T {}\mrightarrow{} \mBbbP{}]. (\mexists{}x:T. (A \mwedge{} B[x]) \mLeftarrow{}{}\mRightarrow{} A \mwedge{} (\mexists{}x:T. B[x]))
Date html generated:
2019_06_20-AM-11_16_30
Last ObjectModification:
2018_09_26-AM-10_01_15
Theory : core_2
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