Nuprl Lemma : predicate_equivalent_implies
∀[T:Type]. ∀[P1,P2:T ⟶ Type]. (P1 ⇐⇒ P2 ⇐⇒ P1 ⇒ P2 ∧ P2 ⇒ P1)
Proof
Definitions occuring in Statement :
predicate_equivalent: P1 ⇐⇒ P2,
predicate_implies: P1 ⇒ P2,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
predicate_implies: P1 ⇒ P2,
predicate_equivalent: P1 ⇐⇒ P2,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
all_wf,
iff_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
applyEquality,
hypothesisEquality,
because_Cache,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
lambdaEquality,
hypothesis,
universeEquality,
productElimination,
functionEquality,
cumulativity,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[P1,P2:T {}\mrightarrow{} Type]. (P1 \mLeftarrow{}{}\mRightarrow{} P2 \mLeftarrow{}{}\mRightarrow{} P1 {}\mRightarrow{} P2 \mwedge{} P2 {}\mRightarrow{} P1)
Date html generated:
2016_05_14-AM-06_05_44
Last ObjectModification:
2015_12_26-AM-11_32_32
Theory : relations
Home
Index