Nuprl Lemma : all_functionality_wrt_uiff
∀[S,T:Type]. ∀[P:S ⟶ ℙ]. ∀[Q:T ⟶ ℙ].
(∀x:T. {uiff(P[x];Q[x])}) ⇒ {uiff(∀x:S. P[x];∀x:T. Q[x])} supposing S = T ∈ Type
Proof
Definitions occuring in Statement :
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
all: ∀x:A. B[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
trivial-equal,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
Error :lambdaFormation_alt,
independent_pairFormation,
Error :functionIsType,
because_Cache,
Error :universeIsType,
applyEquality,
hypothesisEquality,
sqequalHypSubstitution,
Error :productIsType,
Error :isectIsType,
extract_by_obid,
isectElimination,
hyp_replacement,
equalitySymmetry,
Error :equalityIstype,
universeEquality,
Error :inhabitedIsType,
instantiate,
Error :lambdaEquality_alt,
dependent_functionElimination,
functionExtensionality,
productElimination,
equalityTransitivity,
independent_functionElimination
Latex:
\mforall{}[S,T:Type]. \mforall{}[P:S {}\mrightarrow{} \mBbbP{}]. \mforall{}[Q:T {}\mrightarrow{} \mBbbP{}].
(\mforall{}x:T. \{uiff(P[x];Q[x])\}) {}\mRightarrow{} \{uiff(\mforall{}x:S. P[x];\mforall{}x:T. Q[x])\} supposing S = T
Date html generated:
2019_06_20-AM-11_14_48
Last ObjectModification:
2018_11_28-AM-08_53_14
Theory : core_2
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