Nuprl Lemma : decidable__and2
∀[P:ℙ]. ∀[Q:⋂:P. ℙ]. (Dec(P) ⇒ (P ⇒ Dec(Q)) ⇒ Dec(P ∧ Q))
Proof
Definitions occuring in Statement :
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
and: P ∧ Q,
isect: ⋂x:A. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
not: ¬A,
and: P ∧ Q,
false: False
Lemmas referenced :
decidable_wf,
not_wf,
decidable__and
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
thin,
independent_functionElimination,
hypothesis,
cut,
lemma_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
lambdaEquality,
rename,
equalityTransitivity,
equalitySymmetry,
because_Cache,
sqequalRule,
inlFormation,
inrFormation,
productElimination,
voidElimination,
productEquality,
cumulativity,
functionEquality,
isectEquality,
universeEquality
Latex:
\mforall{}[P:\mBbbP{}]. \mforall{}[Q:\mcap{}:P. \mBbbP{}]. (Dec(P) {}\mRightarrow{} (P {}\mRightarrow{} Dec(Q)) {}\mRightarrow{} Dec(P \mwedge{} Q))
Date html generated:
2016_05_13-PM-03_09_02
Last ObjectModification:
2016_01_06-PM-05_27_22
Theory : core_2
Home
Index