Nuprl Lemma : assert_of_eq_bool
∀[p,q:𝔹]. uiff(↑p =b q;p = q)
Proof
Definitions occuring in Statement :
eq_bool: p =b q,
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
implies: P ⇒ Q,
prop: ℙ,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
member: t ∈ T,
uall: ∀[x:A]. B[x],
eq_bool: p =b q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
assert: ↑b,
ifthenelse: if b then t else f fi ,
iff: P ⇐⇒ Q,
true: True,
false: False,
rev_implies: P ⇐ Q,
bor: p ∨bq,
band: p ∧b q,
bnot: ¬bb,
all: ∀x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
exists: ∃x:A. B[x],
not: ¬A,
subtype_rel: A ⊆r B
Lemmas referenced :
bool_wf,
equal_wf,
assert_witness,
eq_bool_wf,
assert_wf,
btrue_wf,
iff_imp_equal_bool,
bfalse_wf,
istype-void,
true_wf,
istype-assert,
bor_wf,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
band_wf,
bnot_wf,
bool_cases_sqequal,
eqff_to_assert,
assert_of_bnot,
btrue_neq_bfalse
Rules used in proof :
Error :universeIsType,
because_Cache,
Error :inhabitedIsType,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
extract_by_obid,
hypothesis,
axiomEquality,
hypothesisEquality,
isectElimination,
isect_memberEquality,
independent_pairEquality,
thin,
productElimination,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
unionElimination,
equalityElimination,
independent_pairFormation,
independent_isectElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
dependent_functionElimination,
instantiate,
cumulativity,
Error :dependent_pairFormation_alt,
Error :equalityIstype,
promote_hyp,
voidElimination,
applyEquality,
baseClosed,
sqequalBase
Latex:
\mforall{}[p,q:\mBbbB{}]. uiff(\muparrow{}p =b q;p = q)
Date html generated:
2019_06_20-AM-11_31_32
Last ObjectModification:
2019_01_06-AM-11_55_40
Theory : bool_1
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