### 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])))`

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` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = 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: `ℙ` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` cand: `A 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: `P `` Q` guard: `{T}` so_lambda: `λ2x y.t[x; y]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` iff: `P `⇐⇒` Q` rev_implies: `P `` 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

Latex:
\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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