Nuprl Lemma : dl-valid-box-test-implies
∀phi,psi:Prop. (|= [(psi)?] phi ⇒ |= psi ⇒ phi)
Proof
Definitions occuring in Statement :
dl-valid: |= phi,
dl-box: [x1] x,
dl-implies: x1 ⇒ x,
dl-test: (x)?,
dl-prop: Prop,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
dl-valid: Error :dl-valid,
member: t ∈ T,
dl-sem: Error :dl-sem,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
and: P ∧ Q,
cand: A c∧ B,
pi2: snd(t),
subtype_rel: A ⊆r B,
prop: ℙ
Lemmas referenced :
istype-void,
istype-atom,
subtype_rel_self,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
isect_memberEquality_alt,
voidElimination,
independent_functionElimination,
independent_pairFormation,
because_Cache,
universeIsType,
applyEquality,
lambdaEquality_alt,
inhabitedIsType,
productElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
functionIsType
Latex:
\mforall{}phi,psi:Prop. (|= [(psi)?] phi {}\mRightarrow{} |= psi {}\mRightarrow{} phi)
Date html generated:
2019_10_15-AM-11_45_26
Last ObjectModification:
2019_03_26-AM-11_28_46
Theory : dynamic!logic
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