Nuprl Lemma : classical-all
∀[T:Type]. ∀[P:T ⟶ ℙ]. (∀x:T. {P[x]} ⇐⇒ {∀x:T. {P[x]}})
Proof
Definitions occuring in Statement :
classical: {P},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
classical: {P},
unit: Unit,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
it_wf,
classical_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
productElimination,
independent_pairEquality,
dependent_functionElimination,
dependent_set_memberEquality,
axiomEquality,
natural_numberEquality,
setElimination,
rename,
universeEquality,
functionEquality,
cumulativity,
isect_memberEquality,
because_Cache
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. (\mforall{}x:T. \{P[x]\} \mLeftarrow{}{}\mRightarrow{} \{\mforall{}x:T. \{P[x]\}\})
Date html generated:
2016_05_13-PM-03_17_06
Last ObjectModification:
2016_01_06-PM-05_21_04
Theory : core_2
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