Nuprl Lemma : decidable__uimplies
∀[P:ℙ]. ∀Q:⋂x:P. ℙ. (Dec(P) ⇒ Dec(Q) supposing P ⇒ Dec(Q supposing P))
Proof
Definitions occuring in Statement :
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
isect: ⋂x:A. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
not: ¬A,
false: False
Lemmas referenced :
isect_wf,
decidable_wf,
isect_subtype_rel_trivial,
subtype_rel_weakening,
ext-eq_weakening,
subtype_rel_wf,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
hypothesisEquality,
lambdaEquality,
applyEquality,
instantiate,
cumulativity,
universeEquality,
independent_isectElimination,
dependent_pairFormation,
because_Cache,
hypothesis,
isectEquality,
unionElimination,
rename,
equalityTransitivity,
equalitySymmetry,
inlFormation,
inrFormation,
independent_functionElimination,
voidElimination
Latex:
\mforall{}[P:\mBbbP{}]. \mforall{}Q:\mcap{}x:P. \mBbbP{}. (Dec(P) {}\mRightarrow{} Dec(Q) supposing P {}\mRightarrow{} Dec(Q supposing P))
Date html generated:
2016_05_15-PM-03_37_48
Last ObjectModification:
2015_12_27-PM-01_15_52
Theory : general
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