Nuprl Lemma : quotient-equality

[T:Type]. ∀[E1,E2:T ⟶ T ⟶ ℙ].
  ((x,y:T//E1[x;y]) (x,y:T//E2[x;y]) ∈ Type) supposing (EquivRel(T;x,y.E1[x;y]) and (∀x,y:T.  (E2[x;y] ⇐⇒ E1[x;y])))


Definitions occuring in Statement :  equiv_rel: EquivRel(T;x,y.E[x; y]) quotient: x,y:A//B[x; y] uimplies: supposing a uall: [x:A]. B[x] prop: so_apply: x[s1;s2] all: x:A. B[x] iff: ⇐⇒ Q function: x:A ⟶ B[x] universe: Type equal: t ∈ T
Definitions unfolded in proof :  quotient: x,y:A//B[x; y] and: P ∧ Q member: t ∈ T uall: [x:A]. B[x] prop: so_apply: x[s1;s2] subtype_rel: A ⊆B cand: c∧ B equiv_rel: EquivRel(T;x,y.E[x; y]) trans: Trans(T;x,y.E[x; y]) all: x:A. B[x] implies:  Q guard: {T} so_lambda: λ2y.t[x; y] so_lambda: λ2x.t[x] so_apply: x[s] uimplies: supposing a iff: ⇐⇒ Q rev_implies:  Q sym: Sym(T;x,y.E[x; y])
Lemmas referenced :  equal-wf-base equiv_rel_wf all_wf iff_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep pertypeEquality productEquality cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality because_Cache hypothesis applyEquality functionExtensionality cumulativity lambdaEquality universeEquality productElimination equalityTransitivity equalitySymmetry independent_pairFormation dependent_functionElimination independent_functionElimination functionEquality isect_memberFormation isect_memberEquality axiomEquality

\mforall{}[T:Type].  \mforall{}[E1,E2:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
    ((x,y:T//E1[x;y])  =  (x,y:T//E2[x;y]))  supposing 
          (EquivRel(T;x,y.E1[x;y])  and 
          (\mforall{}x,y:T.    (E2[x;y]  \mLeftarrow{}{}\mRightarrow{}  E1[x;y])))

Date html generated: 2016_10_21-AM-09_43_41
Last ObjectModification: 2016_08_09-PM-00_18_17

Theory : quot_1

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