### Nuprl Lemma : rel_equivalent_transitivity

`∀[T:Type]. ∀[R1,R2,R3:T ⟶ T ⟶ ℙ].  (R1 `⇐⇒` R2 `` R2 `⇐⇒` R3 `` R1 `⇐⇒` R3)`

Proof

Definitions occuring in Statement :  rel_equivalent: `R1 `⇐⇒` R2` uall: `∀[x:A]. B[x]` prop: `ℙ` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  rel_equivalent: `R1 `⇐⇒` R2` uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` member: `t ∈ T` prop: `ℙ` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` guard: `{T}`
Lemmas referenced :  all_wf iff_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation independent_pairFormation applyEquality hypothesisEquality cut lemma_by_obid sqequalHypSubstitution isectElimination thin lambdaEquality hypothesis functionEquality cumulativity universeEquality dependent_functionElimination productElimination independent_functionElimination

Latex:
\mforall{}[T:Type].  \mforall{}[R1,R2,R3:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    (R1  \mLeftarrow{}{}\mRightarrow{}  R2  {}\mRightarrow{}  R2  \mLeftarrow{}{}\mRightarrow{}  R3  {}\mRightarrow{}  R1  \mLeftarrow{}{}\mRightarrow{}  R3)

Date html generated: 2016_05_14-AM-06_04_42
Last ObjectModification: 2015_12_26-AM-11_33_07

Theory : relations

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