Nuprl Lemma : all_functionality_wrt_uimplies
∀[S,T:Type]. ∀[P:S ⟶ ℙ]. ∀[Q:T ⟶ ℙ].
(∀x:T. {Q[x] supposing P[x]}) ⇒ {∀x:T. Q[x] supposing ∀x:S. P[x]} supposing S = T ∈ Type
Proof
Definitions occuring in Statement :
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 :
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
implies: P ⇒ Q,
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x]
Lemmas referenced :
all_wf,
isect_wf,
equal_wf
Rules used in proof :
functionEquality,
universeEquality,
instantiate,
because_Cache,
equalitySymmetry,
hyp_replacement,
cumulativity,
applyEquality,
lambdaEquality,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
lemma_by_obid,
lambdaFormation,
rename,
thin,
hypothesis,
axiomEquality,
introduction,
cut,
isect_memberFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution,
isectEquality,
equalityTransitivity,
dependent_functionElimination,
functionExtensionality
Latex:
\mforall{}[S,T:Type]. \mforall{}[P:S {}\mrightarrow{} \mBbbP{}]. \mforall{}[Q:T {}\mrightarrow{} \mBbbP{}].
(\mforall{}x:T. \{Q[x] supposing P[x]\}) {}\mRightarrow{} \{\mforall{}x:T. Q[x] supposing \mforall{}x:S. P[x]\} supposing S = T
Date html generated:
2018_05_21-PM-00_00_07
Last ObjectModification:
2018_05_15-PM-04_40_32
Theory : core_2
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