Nuprl Lemma : hereditarily-varterm
∀[opr:Type]. ∀[P:term(opr) ⟶ ℙ]. ∀[v:{v:varname()| ¬(v = nullvar() ∈ varname())} ].
  (hereditarily(opr;s.P[s];varterm(v)) 
⇐⇒ P[varterm(v)])
Proof
Definitions occuring in Statement : 
hereditarily: hereditarily(opr;s.P[s];t)
, 
varterm: varterm(v)
, 
term: term(opr)
, 
nullvar: nullvar()
, 
varname: varname()
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
so_apply: x[s]
, 
iff: P 
⇐⇒ Q
, 
not: ¬A
, 
set: {x:A| B[x]} 
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
member: t ∈ T
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
uimplies: b supposing a
, 
not: ¬A
, 
false: False
, 
prop: ℙ
, 
rev_implies: P 
⇐ Q
, 
hereditarily: hereditarily(opr;s.P[s];t)
, 
all: ∀x:A. B[x]
Lemmas referenced : 
hereditarily_wf, 
term_wf, 
varterm_wf, 
nullvar_wf, 
subterm_wf, 
varname_wf, 
istype-void, 
istype-universe, 
subterm-varterm
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
independent_pairFormation, 
lambdaFormation_alt, 
universeIsType, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
because_Cache, 
sqequalRule, 
lambdaEquality_alt, 
applyEquality, 
hypothesisEquality, 
hypothesis, 
setElimination, 
rename, 
independent_isectElimination, 
independent_functionElimination, 
voidElimination, 
equalityIstype, 
inhabitedIsType, 
setIsType, 
functionIsType, 
universeEquality, 
instantiate, 
productElimination
Latex:
\mforall{}[opr:Type].  \mforall{}[P:term(opr)  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[v:\{v:varname()|  \mneg{}(v  =  nullvar())\}  ].
    (hereditarily(opr;s.P[s];varterm(v))  \mLeftarrow{}{}\mRightarrow{}  P[varterm(v)])
Date html generated:
2020_05_19-PM-09_54_35
Last ObjectModification:
2020_03_12-AM-11_08_47
Theory : terms
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