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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