### Nuprl Lemma : rel_equivalent_weakening

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

Proof

Definitions occuring in Statement :  rel_equivalent: `R1 `⇐⇒` R2` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  rel_equivalent: `R1 `⇐⇒` R2` uall: `∀[x:A]. B[x]` uimplies: `b supposing a` member: `t ∈ T` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` subtype_rel: `A ⊆r B` prop: `ℙ` rev_implies: `P `` Q`
Lemmas referenced :  and_wf equal_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation cut introduction axiomEquality hypothesis thin rename lambdaFormation independent_pairFormation dependent_set_memberEquality hypothesisEquality instantiate extract_by_obid sqequalHypSubstitution isectElimination functionEquality applyEquality lambdaEquality cumulativity universeEquality because_Cache setElimination productElimination setEquality equalityTransitivity equalitySymmetry hyp_replacement Error :applyLambdaEquality,  functionExtensionality

Latex:
\mforall{}[T:Type].  \mforall{}[R1,R2:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    R1  \mLeftarrow{}{}\mRightarrow{}  R2  supposing  R1  =  R2

Date html generated: 2016_10_21-AM-09_43_33
Last ObjectModification: 2016_07_12-AM-05_04_09

Theory : relations

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