Nuprl Lemma : mFOL-subst-trivial
∀nw,old:ℤ. ∀fmla:mFOL().
  ((¬(old ∈ mFOL-freevars(fmla)))
  
⇒ (mFOL-abstract(fmla[nw/old]) = mFOL-abstract(fmla) ∈ AbstractFOFormula(mFOL-freevars(fmla))))
Proof
Definitions occuring in Statement : 
mFOL-subst: fmla[nw/old]
, 
mFOL-abstract: mFOL-abstract(fmla)
, 
mFOL-freevars: mFOL-freevars(fmla)
, 
mFOL: mFOL()
, 
AbstractFOFormula: AbstractFOFormula(vs)
, 
l_member: (x ∈ l)
, 
all: ∀x:A. B[x]
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
int: ℤ
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
mFOL-subst: fmla[nw/old]
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
subtype_rel: A ⊆r B
, 
so_lambda: λ2x.t[x]
, 
and: P ∧ Q
, 
prop: ℙ
, 
uimplies: b supposing a
, 
squash: ↓T
, 
true: True
, 
so_apply: x[s]
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
mFOL-rename-bound-to-avoid_wf, 
cons_wf, 
nil_wf, 
set_wf, 
mFOL_wf, 
equal_wf, 
list_wf, 
mFOL-freevars_wf, 
length_wf, 
mFOL-abstract_wf, 
subtype_rel-equal, 
AbstractFOFormula_wf, 
l_disjoint_wf, 
mFOL-boundvars_wf, 
trivial-mFOL-rename, 
squash_wf, 
true_wf, 
not_wf, 
l_member_wf, 
iff_weakening_equal, 
mFOL-rename_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
dependent_functionElimination, 
thin, 
hypothesisEquality, 
isectElimination, 
intEquality, 
hypothesis, 
instantiate, 
applyEquality, 
lambdaEquality, 
cumulativity, 
universeEquality, 
sqequalRule, 
productEquality, 
applyLambdaEquality, 
because_Cache, 
independent_isectElimination, 
imageElimination, 
equalitySymmetry, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
setElimination, 
rename, 
productElimination, 
equalityTransitivity, 
equalityUniverse, 
levelHypothesis, 
independent_functionElimination
Latex:
\mforall{}nw,old:\mBbbZ{}.  \mforall{}fmla:mFOL().
    ((\mneg{}(old  \mmember{}  mFOL-freevars(fmla)))  {}\mRightarrow{}  (mFOL-abstract(fmla[nw/old])  =  mFOL-abstract(fmla)))
Date html generated:
2018_05_21-PM-10_24_23
Last ObjectModification:
2017_07_26-PM-06_38_30
Theory : minimal-first-order-logic
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