Nuprl Lemma : bimplies_transitivity
∀[u,v,w:𝔹]. (↑(u ⇒b w)) supposing ((↑(v ⇒b w)) and (↑(u ⇒b v)))
Proof
Definitions occuring in Statement :
bimplies: p ⇒b q,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
bimplies: p ⇒b q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bor: p ∨bq,
bfalse: ff,
assert: ↑b,
prop: ℙ,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
false: False,
true: True
Lemmas referenced :
bool_wf,
eqtt_to_assert,
assert_witness,
assert_wf,
true_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert_of_bnot,
false_wf,
bor_wf,
bnot_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
thin,
extract_by_obid,
hypothesis,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
equalityElimination,
isectElimination,
productElimination,
independent_isectElimination,
because_Cache,
independent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
voidElimination,
natural_numberEquality,
axiomEquality
Latex:
\mforall{}[u,v,w:\mBbbB{}]. (\muparrow{}(u {}\mRightarrow{}\msubb{} w)) supposing ((\muparrow{}(v {}\mRightarrow{}\msubb{} w)) and (\muparrow{}(u {}\mRightarrow{}\msubb{} v)))
Date html generated:
2019_06_20-AM-11_31_22
Last ObjectModification:
2018_08_27-PM-03_39_42
Theory : bool_1
Home
Index