Nuprl Lemma : freevars-FOL-subst
∀fmla:mFOL(). ∀nw,old,x:ℤ.
  ((x ∈ mFOL-freevars(fmla[nw/old]))
  
⇐⇒ ((¬(x = old ∈ ℤ)) ∧ (x ∈ mFOL-freevars(fmla))) ∨ ((x = nw ∈ ℤ) ∧ (old ∈ mFOL-freevars(fmla))))
Proof
Definitions occuring in Statement : 
FOL-subst: fmla[nw/old]
, 
mFOL-freevars: mFOL-freevars(fmla)
, 
mFOL: mFOL()
, 
l_member: (x ∈ l)
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
not: ¬A
, 
or: P ∨ Q
, 
and: P ∧ Q
, 
int: ℤ
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
FOL-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]
, 
implies: P 
⇒ Q
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
or: P ∨ Q
, 
not: ¬A
, 
false: False
, 
l_disjoint: l_disjoint(T;l1;l2)
, 
cand: A c∧ B
Lemmas referenced : 
FOL-rename-bound-to-avoid_wf, 
cons_wf, 
nil_wf, 
set_wf, 
mFOL_wf, 
equal_wf, 
list_wf, 
mFOL-freevars_wf, 
length_wf, 
FOL-abstract_wf, 
subtype_rel-equal, 
AbstractFOFormula+_wf, 
l_disjoint_wf, 
mFOL-boundvars_wf, 
and_wf, 
l_member_wf, 
not_wf, 
or_wf, 
equal-wf-base, 
int_subtype_base, 
mFOL-freevars-of-rename, 
mFOL-rename_wf, 
iff_wf, 
cons_member
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, 
independent_pairFormation, 
addLevel, 
hyp_replacement, 
setElimination, 
rename, 
productElimination, 
levelHypothesis, 
dependent_set_memberEquality, 
equalityTransitivity, 
impliesFunctionality, 
independent_functionElimination, 
inlFormation
Latex:
\mforall{}fmla:mFOL().  \mforall{}nw,old,x:\mBbbZ{}.
    ((x  \mmember{}  mFOL-freevars(fmla[nw/old]))
    \mLeftarrow{}{}\mRightarrow{}  ((\mneg{}(x  =  old))  \mwedge{}  (x  \mmember{}  mFOL-freevars(fmla)))  \mvee{}  ((x  =  nw)  \mwedge{}  (old  \mmember{}  mFOL-freevars(fmla))))
Date html generated:
2018_05_21-PM-10_24_34
Last ObjectModification:
2017_07_26-PM-06_38_35
Theory : minimal-first-order-logic
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