Nuprl Lemma : equal_bool_if
∀[a,b:𝔹]. (a = b) supposing ((¬((↑a) ∧ (¬↑b))) and (¬((¬↑a) ∧ (↑b))))
Proof
Definitions occuring in Statement :
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
bool: 𝔹,
unit: Unit,
member: t ∈ T,
it: ⋅,
btrue: tt,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
false: False,
not: ¬A,
cand: A c∧ B,
true: True,
prop: ℙ,
rev_implies: P ⇐ Q
Lemmas referenced :
btrue_wf,
iff_imp_equal_bool,
bfalse_wf,
istype-void,
true_wf,
assert_wf,
bool_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
sqequalHypSubstitution,
unionElimination,
thin,
equalityElimination,
sqequalRule,
cut,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
independent_isectElimination,
independent_pairFormation,
Error :lambdaFormation_alt,
independent_functionElimination,
natural_numberEquality,
voidElimination,
Error :universeIsType,
Error :functionIsType,
Error :productIsType,
because_Cache,
hypothesisEquality,
Error :inhabitedIsType
Latex:
\mforall{}[a,b:\mBbbB{}]. (a = b) supposing ((\mneg{}((\muparrow{}a) \mwedge{} (\mneg{}\muparrow{}b))) and (\mneg{}((\mneg{}\muparrow{}a) \mwedge{} (\muparrow{}b))))
Date html generated:
2019_06_20-AM-11_31_27
Last ObjectModification:
2018_10_10-PM-06_41_47
Theory : bool_1
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