### Nuprl Lemma : decidable__cand

`∀[P:ℙ]. ∀[Q:⋂x:P. ℙ].  (Dec(P) `` (P `` Dec(Q)) `` Dec(P c∧ Q))`

Proof

Definitions occuring in Statement :  decidable: `Dec(P)` uall: `∀[x:A]. B[x]` cand: `A c∧ B` prop: `ℙ` implies: `P `` Q` isect: `⋂x:A. B[x]`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` cand: `A c∧ B` and: `P ∧ Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` exists: `∃x:A. B[x]`
Lemmas referenced :  decidable__and2 decidable_wf isect_subtype_rel_trivial subtype_rel_weakening ext-eq_weakening subtype_rel_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality independent_functionElimination hypothesis because_Cache functionEquality applyEquality instantiate cumulativity universeEquality sqequalRule lambdaEquality independent_isectElimination dependent_pairFormation isectIsType universeIsType inhabitedIsType

Latex:
\mforall{}[P:\mBbbP{}].  \mforall{}[Q:\mcap{}x:P.  \mBbbP{}].    (Dec(P)  {}\mRightarrow{}  (P  {}\mRightarrow{}  Dec(Q))  {}\mRightarrow{}  Dec(P  c\mwedge{}  Q))

Date html generated: 2019_10_15-AM-10_47_01
Last ObjectModification: 2018_09_27-AM-09_35_29

Theory : basic

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