Nuprl Lemma : prob2.4
∀[U:Type]. ∀[P,Q:U ⟶ ℙ]. ((∃x:U. ((P x) ∧ (Q x))) ⇒ ((∃x:U. (P x)) ∧ (∃x:U. (Q x))))
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
prop: ℙ,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
exists_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
hypothesis,
applyEquality,
independent_pairFormation,
because_Cache,
lemma_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[U:Type]. \mforall{}[P,Q:U {}\mrightarrow{} \mBbbP{}]. ((\mexists{}x:U. ((P x) \mwedge{} (Q x))) {}\mRightarrow{} ((\mexists{}x:U. (P x)) \mwedge{} (\mexists{}x:U. (Q x))))
Date html generated:
2016_05_15-PM-07_43_51
Last ObjectModification:
2015_12_27-AM-11_11_29
Theory : general
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