Nuprl Lemma : ifthenelse_functionality_wrt_iff
∀b1,b2:𝔹.
∀[p1,q1,p2,q2:ℙ].
(b1 = b2 ⇒ {q1 ⇐⇒ q2} ⇒ {p1 ⇐⇒ p2} ⇒ {if b1 then p1 else q1 fi ⇐⇒ if b2 then p2 else q2 fi })
Proof
Definitions occuring in Statement :
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
assert: ↑b,
true: True,
prop: ℙ,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
false: False
Lemmas referenced :
eqtt_to_assert,
subtype_base_sq,
bool_subtype_base,
iff_imp_equal_bool,
btrue_wf,
assert_wf,
true_wf,
eqff_to_assert,
equal_wf,
bool_wf,
bool_cases_sqequal,
assert_of_bnot,
ifthenelse_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
cut,
hypothesisEquality,
because_Cache,
unionElimination,
equalityElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
independent_isectElimination,
independent_functionElimination,
instantiate,
natural_numberEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
voidElimination,
cumulativity,
universeEquality
Latex:
\mforall{}b1,b2:\mBbbB{}.
\mforall{}[p1,q1,p2,q2:\mBbbP{}].
(b1 = b2
{}\mRightarrow{} \{q1 \mLeftarrow{}{}\mRightarrow{} q2\}
{}\mRightarrow{} \{p1 \mLeftarrow{}{}\mRightarrow{} p2\}
{}\mRightarrow{} \{if b1 then p1 else q1 fi \mLeftarrow{}{}\mRightarrow{} if b2 then p2 else q2 fi \})
Date html generated:
2017_04_14-AM-07_29_57
Last ObjectModification:
2017_02_27-PM-02_58_38
Theory : bool_1
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