Nuprl Lemma : exists-simp
∀[T:Type]. ∀a:T. ∀P:T ⟶ ℙ. ((∀x:T. (P[x] ⇒ (x = a ∈ T))) ⇒ (∃x:T. P[x] ⇐⇒ P[a]))
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q
Lemmas referenced :
and_wf,
equal_wf,
exists_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
dependent_set_memberEquality,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
sqequalRule,
cumulativity,
functionExtensionality,
dependent_pairFormation,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}a:T. \mforall{}P:T {}\mrightarrow{} \mBbbP{}. ((\mforall{}x:T. (P[x] {}\mRightarrow{} (x = a))) {}\mRightarrow{} (\mexists{}x:T. P[x] \mLeftarrow{}{}\mRightarrow{} P[a]))
Date html generated:
2016_10_25-AM-10_43_29
Last ObjectModification:
2016_07_12-AM-06_53_46
Theory : general
Home
Index