Nuprl Lemma : half-squash-stable__all
∀[T:Type]. ∀[P:T ⟶ ℙ]. ((∀x:T. half-squash-stable(P[x])) ⇒ half-squash-stable(∀x:T. P[x]))
Proof
Definitions occuring in Statement :
half-squash-stable: half-squash-stable(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
member: t ∈ T,
all: ∀x:A. B[x],
half-squash-stable: half-squash-stable(P),
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
implies-quotient-true,
half-squash-stable_wf,
equiv_rel_true,
true_wf,
all_wf,
quotient_wf
Rules used in proof :
dependent_functionElimination,
promote_hyp,
independent_functionElimination,
universeEquality,
functionEquality,
independent_isectElimination,
hypothesis,
because_Cache,
functionExtensionality,
applyEquality,
lambdaEquality,
sqequalRule,
cumulativity,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
hypothesisEquality,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x:T. half-squash-stable(P[x])) {}\mRightarrow{} half-squash-stable(\mforall{}x:T. P[x]))
Date html generated:
2017_09_29-PM-05_48_11
Last ObjectModification:
2017_08_30-AM-11_09_48
Theory : quot_1
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