Nuprl Lemma : test-exists-elim
∀[A:Type]. ∀[Q:A ⟶ A ⟶ ℙ]. ∀x,b:A. (∃z:A × A. (Q[fst(z);snd(z)] ∧ (z = <x, b> ∈ (A × A))) ⇐⇒ Q[x;b] ∧ (x = x ∈ A))
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
pi1: fst(t),
pi2: snd(t),
rev_implies: P ⇐ Q
Lemmas referenced :
and_wf,
equal_wf,
pi1_wf,
pi2_wf,
subtype_rel_self,
exists_wf,
member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
hypothesisEquality,
functionEquality,
cumulativity,
universeEquality,
cut,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
independent_functionElimination,
dependent_pairFormation,
because_Cache,
hypothesis,
hyp_replacement,
equalitySymmetry,
sqequalRule,
dependent_set_memberEquality,
introduction,
extract_by_obid,
isectElimination,
productEquality,
equalityTransitivity,
applyLambdaEquality,
setElimination,
rename,
lambdaEquality,
applyEquality,
independent_pairEquality,
dependent_functionElimination,
instantiate,
promote_hyp
Latex:
\mforall{}[A:Type]. \mforall{}[Q:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
\mforall{}x,b:A. (\mexists{}z:A \mtimes{} A. (Q[fst(z);snd(z)] \mwedge{} (z = <x, b>)) \mLeftarrow{}{}\mRightarrow{} Q[x;b] \mwedge{} (x = x))
Date html generated:
2019_10_15-AM-11_06_52
Last ObjectModification:
2018_09_25-PM-00_08_23
Theory : general
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