### Nuprl Lemma : quotient_wf

`∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ].  x,y:T//E[x;y] ∈ Type supposing 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` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` member: `t ∈ T` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` and: `P ∧ Q` subtype_rel: `A ⊆r B` cand: `A c∧ B` guard: `{T}` 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` quotient: `x,y:A//B[x; y]` sym: `Sym(T;x,y.E[x; y])`
Lemmas referenced :  equiv_rel_wf equal-wf-base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution hypothesis sqequalRule axiomEquality equalityTransitivity equalitySymmetry lemma_by_obid isectElimination thin hypothesisEquality lambdaEquality applyEquality isect_memberEquality because_Cache functionEquality cumulativity universeEquality productEquality productElimination independent_pairFormation dependent_functionElimination independent_functionElimination pertypeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[E:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    x,y:T//E[x;y]  \mmember{}  Type  supposing  EquivRel(T;x,y.E[x;y])

Date html generated: 2016_05_14-AM-06_07_42
Last ObjectModification: 2015_12_26-AM-11_48_38

Theory : quot_1

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