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Step * 3 of Lemma mFOL-bound-rename


1. isall : 𝔹
2. var : ℤ
3. body : mFOL()
4. ∀L:ℤ List
     (∃fmla':mFOL() [((mFOL-freevars(fmla') = mFOL-freevars(body) ∈ (ℤ List))
                    ∧ (mFOL-abstract(fmla') = mFOL-abstract(body) ∈ AbstractFOFormula(mFOL-freevars(body)))
                    ∧ l_disjoint(ℤ;L;mFOL-boundvars(fmla')))])
5. L : ℤ List
⊢ ∃fmla':mFOL() [((mFOL-freevars(fmla') = mFOL-freevars(mFOquant(isall;var;body)) ∈ (ℤ List))
                ∧ (mFOL-abstract(fmla')
                  = mFOL-abstract(mFOquant(isall;var;body))
                  ∈ AbstractFOFormula(mFOL-freevars(mFOquant(isall;var;body))))
                ∧ l_disjoint(ℤ;L;mFOL-boundvars(fmla')))]
BY
{ (RenameVar `z' 2 THEN Decide (z ∈ L) THEN Auto) }

1
1. isall : 𝔹
2. z : ℤ
3. body : mFOL()
4. ∀L:ℤ List
     (∃fmla':mFOL() [((mFOL-freevars(fmla') = mFOL-freevars(body) ∈ (ℤ List))
                    ∧ (mFOL-abstract(fmla') = mFOL-abstract(body) ∈ AbstractFOFormula(mFOL-freevars(body)))
                    ∧ l_disjoint(ℤ;L;mFOL-boundvars(fmla')))])
5. L : ℤ List
6. (z ∈ L)
⊢ ∃fmla':mFOL() [((mFOL-freevars(fmla') = mFOL-freevars(mFOquant(isall;z;body)) ∈ (ℤ List))
                ∧ (mFOL-abstract(fmla')
                  = mFOL-abstract(mFOquant(isall;z;body))
                  ∈ AbstractFOFormula(mFOL-freevars(mFOquant(isall;z;body))))
                ∧ l_disjoint(ℤ;L;mFOL-boundvars(fmla')))]

2
1. isall : 𝔹
2. z : ℤ
3. body : mFOL()
4. ∀L:ℤ List
     (∃fmla':mFOL() [((mFOL-freevars(fmla') = mFOL-freevars(body) ∈ (ℤ List))
                    ∧ (mFOL-abstract(fmla') = mFOL-abstract(body) ∈ AbstractFOFormula(mFOL-freevars(body)))
                    ∧ l_disjoint(ℤ;L;mFOL-boundvars(fmla')))])
5. L : ℤ List
6. ¬(z ∈ L)
⊢ ∃fmla':mFOL() [((mFOL-freevars(fmla') = mFOL-freevars(mFOquant(isall;z;body)) ∈ (ℤ List))
                ∧ (mFOL-abstract(fmla')
                  = mFOL-abstract(mFOquant(isall;z;body))
                  ∈ AbstractFOFormula(mFOL-freevars(mFOquant(isall;z;body))))
                ∧ l_disjoint(ℤ;L;mFOL-boundvars(fmla')))]

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Latex:

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Latex:

1.  isall  :  \mBbbB{}
2.  var  :  \mBbbZ{}
3.  body  :  mFOL()
4.  \mforall{}L:\mBbbZ{}  List
          (\mexists{}fmla':mFOL()  [((mFOL-freevars(fmla')  =  mFOL-freevars(body))
                                        \mwedge{}  (mFOL-abstract(fmla')  =  mFOL-abstract(body))
                                        \mwedge{}  l\_disjoint(\mBbbZ{};L;mFOL-boundvars(fmla')))])
5.  L  :  \mBbbZ{}  List
\mvdash{}  \mexists{}fmla':mFOL()  [((mFOL-freevars(fmla')  =  mFOL-freevars(mFOquant(isall;var;body)))
                                \mwedge{}  (mFOL-abstract(fmla')  =  mFOL-abstract(mFOquant(isall;var;body)))
                                \mwedge{}  l\_disjoint(\mBbbZ{};L;mFOL-boundvars(fmla')))]


By

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Latex:
(RenameVar  `z'  2  THEN  Decide  (z  \mmember{}  L)  THEN  Auto)

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