Nuprl Lemma : ifthenelse-prop
∀b:𝔹. ∀p,q:ℙ'. (if b then p else q fi ⇐⇒ ((↑b) ∧ p) ∨ ((¬↑b) ∧ q))
Proof
Definitions occuring in Statement :
assert: ↑b,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q,
and: P ∧ Q
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,
or: P ∨ Q,
cand: A c∧ B,
prop: ℙ,
rev_implies: P ⇐ Q,
not: ¬A,
true: True,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb
Lemmas referenced :
bool_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesisEquality,
thin,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
unionElimination,
equalityElimination,
isectElimination,
productElimination,
independent_isectElimination,
sqequalRule,
independent_pairFormation,
inlFormation,
because_Cache,
independent_functionElimination,
natural_numberEquality,
voidElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
inrFormation
Latex:
\mforall{}b:\mBbbB{}. \mforall{}p,q:\mBbbP{}'. (if b then p else q fi \mLeftarrow{}{}\mRightarrow{} ((\muparrow{}b) \mwedge{} p) \mvee{} ((\mneg{}\muparrow{}b) \mwedge{} q))
Date html generated:
2017_04_14-AM-07_32_05
Last ObjectModification:
2017_02_27-PM-02_59_59
Theory : bool_1
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