Nuprl Lemma : xxrefl_functionality_wrt_breqv
∀[T:Type]. ∀[R,R':T ⟶ T ⟶ ℙ]. ((R <≡>{T} R') ⇒ (refl(T;R) ⇐⇒ refl(T;R')))
Proof
Definitions occuring in Statement :
xxrefl: refl(T;E),
binrel_eqv: E <≡>{T} E',
uall: ∀[x:A]. B[x],
prop: ℙ,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
xxrefl: refl(T;E),
binrel_eqv: E <≡>{T} E',
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
all: ∀x:A. B[x]
Lemmas referenced :
all_wf,
iff_wf,
refl_wf,
refl_functionality_wrt_iff
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
independent_pairFormation,
because_Cache,
addLevel,
productElimination,
impliesFunctionality,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R,R':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. ((R <\mequiv{}>\{T\} R') {}\mRightarrow{} (refl(T;R) \mLeftarrow{}{}\mRightarrow{} refl(T;R')))
Date html generated:
2016_05_15-PM-00_00_47
Last ObjectModification:
2015_12_26-PM-11_26_43
Theory : gen_algebra_1
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