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