Nuprl Lemma : rel_implies_weakening
∀[T:Type]. ∀[R1,R2:T ⟶ T ⟶ ℙ]. (R1 ⇐⇒ R2 ⇒ R1 => R2)
Proof
Definitions occuring in Statement :
rel_equivalent: R1 ⇐⇒ R2,
rel_implies: R1 => R2,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
rel_implies: R1 => R2,
rel_equivalent: R1 ⇐⇒ R2,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
infix_ap: x f y,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
all_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
applyEquality,
lemma_by_obid,
isectElimination,
lambdaEquality,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R1,R2:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. (R1 \mLeftarrow{}{}\mRightarrow{} R2 {}\mRightarrow{} R1 => R2)
Date html generated:
2016_05_14-AM-06_04_39
Last ObjectModification:
2015_12_26-AM-11_33_09
Theory : relations
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