Nuprl Lemma : respects-equality-quotient1
∀[X,T:Type]. ∀[E:T ⟶ T ⟶ ℙ].
  (respects-equality(X;x,y:T//E[x;y])) supposing (respects-equality(X;T) and EquivRel(T;x,y.E[x;y]))
Proof
Definitions occuring in Statement : 
equiv_rel: EquivRel(T;x,y.E[x; y])
, 
quotient: x,y:A//B[x; y]
, 
uimplies: b supposing a
, 
respects-equality: respects-equality(S;T)
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
so_apply: x[s1;s2]
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
uimplies: b supposing a
, 
respects-equality: respects-equality(S;T)
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
member: t ∈ T
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
prop: ℙ
, 
quotient: x,y:A//B[x; y]
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
subtype_rel: A ⊆r B
Lemmas referenced : 
quotient_wf, 
istype-base, 
respects-equality_wf, 
equiv_rel_wf, 
istype-universe, 
subtype_rel_self, 
change-equality-type, 
subtype_quotient
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :isect_memberFormation_alt, 
Error :lambdaFormation_alt, 
sqequalRule, 
Error :equalityIstype, 
Error :universeIsType, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
Error :lambdaEquality_alt, 
applyEquality, 
Error :inhabitedIsType, 
independent_isectElimination, 
hypothesis, 
because_Cache, 
sqequalBase, 
equalitySymmetry, 
Error :functionIsType, 
universeEquality, 
instantiate, 
pertypeElimination, 
promote_hyp, 
productElimination, 
Error :productIsType, 
equalityTransitivity, 
independent_functionElimination, 
dependent_functionElimination
Latex:
\mforall{}[X,T:Type].  \mforall{}[E:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
    (respects-equality(X;x,y:T//E[x;y]))  supposing  (respects-equality(X;T)  and  EquivRel(T;x,y.E[x;y]))
Date html generated:
2019_06_20-PM-00_32_26
Last ObjectModification:
2018_12_13-PM-04_09_24
Theory : quot_1
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