Nuprl Lemma : uall_functionality_wrt_iff
∀[S,T:Type]. ∀[P,Q:S ⟶ ℙ]. (∀[x:S]. (P[x] ⇐⇒ Q[x])) ⇒ {∀[x:S]. P[x] ⇐⇒ ∀[y:T]. Q[y]} 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],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
and: P ∧ Q
Lemmas referenced :
equal_wf,
and_wf,
uall_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
hyp_replacement,
hypothesisEquality,
equalitySymmetry,
hypothesis,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
lambdaEquality,
applyEquality,
cumulativity,
because_Cache,
functionEquality,
universeEquality,
instantiate,
isect_memberFormation,
introduction,
axiomEquality,
rename,
lambdaFormation,
independent_pairFormation,
productElimination,
independent_functionElimination
Latex:
\mforall{}[S,T:Type]. \mforall{}[P,Q:S {}\mrightarrow{} \mBbbP{}].
(\mforall{}[x:S]. (P[x] \mLeftarrow{}{}\mRightarrow{} Q[x])) {}\mRightarrow{} \{\mforall{}[x:S]. P[x] \mLeftarrow{}{}\mRightarrow{} \mforall{}[y:T]. Q[y]\} supposing S = T
Date html generated:
2016_05_13-PM-03_07_43
Last ObjectModification:
2016_01_06-PM-05_28_05
Theory : core_2
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