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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