Nuprl Lemma : binrel_eqv_functionality_wrt_breqv
∀[T:Type]. ∀[a,a',b,b':T ⟶ T ⟶ ℙ]. ((a <≡>{T} b) ⇒ (a' <≡>{T} b') ⇒ (a <≡>{T} a' ⇐⇒ b <≡>{T} b'))
Proof
Definitions occuring in Statement :
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
implies: P ⇒ Q,
guard: {T},
uequiv_rel: UniformEquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
urefl: UniformlyRefl(T;x,y.E[x; y]),
usym: UniformlySym(T;x,y.E[x; y]),
utrans: UniformlyTrans(T;x,y.E[x; y]),
uimplies: b supposing a
Lemmas referenced :
uequiv_rel_self_functionality,
binrel_eqv_wf,
binrel_eqv_weakening,
binrel_eqv_inversion,
binrel_eqv_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
thin,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
functionEquality,
cumulativity,
hypothesisEquality,
universeEquality,
sqequalRule,
lambdaEquality,
hypothesis,
independent_functionElimination,
independent_pairFormation,
lambdaFormation,
because_Cache,
independent_isectElimination
Latex:
\mforall{}[T:Type]. \mforall{}[a,a',b,b':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((a <\mequiv{}>\{T\} b) {}\mRightarrow{} (a' <\mequiv{}>\{T\} b') {}\mRightarrow{} (a <\mequiv{}>\{T\} a' \mLeftarrow{}{}\mRightarrow{} b <\mequiv{}>\{T\} b'))
Date html generated:
2016_05_14-PM-03_54_47
Last ObjectModification:
2015_12_26-PM-06_56_01
Theory : relations2
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