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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