Nuprl Lemma : ite_false
∀b:𝔹. ∀[x:ℙ]. (if b then x else False fi ⇐⇒ (↑b) ∧ x)
Proof
Definitions occuring in Statement :
assert: ↑b,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
false: False
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
true: True,
prop: ℙ,
rev_implies: P ⇐ Q,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
false: False
Lemmas referenced :
bool_wf,
eqtt_to_assert,
true_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesisEquality,
thin,
introduction,
extract_by_obid,
hypothesis,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
equalityElimination,
isectElimination,
productElimination,
independent_isectElimination,
sqequalRule,
isect_memberFormation_alt,
independent_pairFormation,
natural_numberEquality,
productEquality,
cumulativity,
universeIsType,
universeEquality,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
voidElimination
Latex:
\mforall{}b:\mBbbB{}. \mforall{}[x:\mBbbP{}]. (if b then x else False fi \mLeftarrow{}{}\mRightarrow{} (\muparrow{}b) \mwedge{} x)
Date html generated:
2019_10_15-AM-10_46_33
Last ObjectModification:
2018_09_27-AM-09_41_14
Theory : basic
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