### Nuprl Lemma : equiv_rel_functionality_wrt_iff

`∀[T,T':Type]. ∀[E:T ⟶ T ⟶ ℙ]. ∀[E':T' ⟶ T' ⟶ ℙ].`
`  (∀x,y:T.  (E[x;y] `⇐⇒` E'[x;y])) `` (EquivRel(T;x,y.E[x;y]) `⇐⇒` EquivRel(T';x,y.E'[x;y])) supposing T = T' ∈ Type`

Proof

Definitions occuring in Statement :  equiv_rel: `EquivRel(T;x,y.E[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` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` uimplies: `b supposing a` member: `t ∈ T` implies: `P `` Q` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` guard: `{T}` so_apply: `x[s]` refl: `Refl(T;x,y.E[x; y])` sym: `Sym(T;x,y.E[x; y])` trans: `Trans(T;x,y.E[x; y])` equiv_rel: `EquivRel(T;x,y.E[x; y])` iff: `P `⇐⇒` Q` and: `P ∧ Q` all: `∀x:A. B[x]` rev_implies: `P `` Q`
Lemmas referenced :  equal_wf ext-eq_weakening subtype_rel_weakening iff_wf all_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation cut introduction axiomEquality hypothesis thin rename lambdaFormation lemma_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality applyEquality independent_isectElimination because_Cache instantiate universeEquality functionEquality cumulativity productElimination productEquality addLevel independent_pairFormation impliesFunctionality allFunctionality dependent_functionElimination independent_functionElimination andLevelFunctionality allLevelFunctionality impliesLevelFunctionality equalitySymmetry

Latex:
\mforall{}[T,T':Type].  \mforall{}[E:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[E':T'  {}\mrightarrow{}  T'  {}\mrightarrow{}  \mBbbP{}].
(\mforall{}x,y:T.    (E[x;y]  \mLeftarrow{}{}\mRightarrow{}  E'[x;y]))  {}\mRightarrow{}  (EquivRel(T;x,y.E[x;y])  \mLeftarrow{}{}\mRightarrow{}  EquivRel(T';x,y.E'[x;y]))
supposing  T  =  T'

Date html generated: 2016_05_13-PM-04_15_23
Last ObjectModification: 2016_01_05-PM-01_47_39

Theory : rel_1

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