Nuprl Lemma : assert_of_band
∀p:𝔹. ∀q:⋂:↑p. 𝔹. uiff(↑(p ∧b q);(↑p) ∧ (↑q))
Proof
Definitions occuring in Statement :
band: p ∧b q,
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
and: P ∧ Q,
isect: ⋂x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
member: t ∈ T,
it: ⋅,
btrue: tt,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
assert: ↑b,
ifthenelse: if b then t else f fi ,
band: p ∧b q,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
true: True
Lemmas referenced :
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
true_wf,
it_wf,
subtype_rel_self,
assert_witness,
assert_wf,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
Error :inhabitedIsType,
hypothesis,
thin,
sqequalHypSubstitution,
unionElimination,
equalityElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
productElimination,
independent_isectElimination,
sqequalRule,
Error :dependent_pairFormation_alt,
Error :equalityIsType1,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
voidElimination,
Error :universeIsType,
isectEquality,
lambdaFormation,
applyEquality,
independent_pairFormation,
Error :isect_memberFormation_alt,
natural_numberEquality,
independent_pairEquality,
axiomEquality,
Error :productIsType,
isect_memberFormation,
productEquality
Latex:
\mforall{}p:\mBbbB{}. \mforall{}q:\mcap{}:\muparrow{}p. \mBbbB{}. uiff(\muparrow{}(p \mwedge{}\msubb{} q);(\muparrow{}p) \mwedge{} (\muparrow{}q))
Date html generated:
2019_06_20-AM-11_31_35
Last ObjectModification:
2018_09_28-PM-11_33_03
Theory : bool_1
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