Nuprl Lemma : exists-simp-test
∀T:Type. ∀P:T ⟶ ℙ'. ∀a:T. (∃x:T. (P[x] ∧ (x = a ∈ T)) ⇐⇒ P[a])
Proof
Definitions occuring in Statement :
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
prop: ℙ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q
Lemmas referenced :
and_wf,
equal_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
addLevel,
hyp_replacement,
equalitySymmetry,
sqequalRule,
dependent_set_memberEquality,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
applyLambdaEquality,
setElimination,
rename,
applyEquality,
levelHypothesis,
instantiate,
cumulativity,
lambdaEquality,
productEquality,
functionExtensionality,
universeEquality,
dependent_pairFormation,
because_Cache,
functionEquality
Latex:
\mforall{}T:Type. \mforall{}P:T {}\mrightarrow{} \mBbbP{}'. \mforall{}a:T. (\mexists{}x:T. (P[x] \mwedge{} (x = a)) \mLeftarrow{}{}\mRightarrow{} P[a])
Date html generated:
2017_10_01-AM-09_10_43
Last ObjectModification:
2017_07_26-PM-04_47_02
Theory : general
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