Nuprl Lemma : FOL-subst-trivial
∀nw,old:ℤ. ∀fmla:mFOL().
((¬(old ∈ mFOL-freevars(fmla)))
⇒ (FOL-abstract(fmla[nw/old]) = FOL-abstract(fmla) ∈ AbstractFOFormula+(mFOL-freevars(fmla))))
Proof
Definitions occuring in Statement :
FOL-subst: fmla[nw/old],
FOL-abstract: FOL-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,
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],
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
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,
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{} (FOL-abstract(fmla[nw/old]) = FOL-abstract(fmla)))
Date html generated:
2018_05_21-PM-10_24_43
Last ObjectModification:
2017_07_26-PM-06_38_39
Theory : minimal-first-order-logic
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