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Step * 3 2 2 1 of Lemma mFOL-bound-rename


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)
7. fmla' : mFOL()
8. mFOL-freevars(fmla') = mFOL-freevars(body) ∈ (ℤ List)
9. mFOL-abstract(fmla') = mFOL-abstract(body) ∈ AbstractFOFormula(mFOL-freevars(body))
10. l_disjoint(ℤ;L;mFOL-boundvars(fmla'))
11. mFOL-freevars(mFOquant(isall;z;fmla')) = mFOL-freevars(mFOquant(isall;z;body)) ∈ (ℤ List)
⊢ (FOQuantifier(isall) z mFOL-abstract(fmla'))
= (FOQuantifier(isall) z mFOL-abstract(body))
∈ AbstractFOFormula(filter(λx.(¬b(x =z z));mFOL-freevars(body)))
BY
{ (InstLemma `FOQuantifier_wf` [⌜mFOL-freevars(fmla')⌝;⌜isall⌝]⋅ THEN Auto) }

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Latex:

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Latex:

1.  isall  :  \mBbbB{}
2.  z  :  \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
6.  \mneg{}(z  \mmember{}  L)
7.  fmla'  :  mFOL()
8.  mFOL-freevars(fmla')  =  mFOL-freevars(body)
9.  mFOL-abstract(fmla')  =  mFOL-abstract(body)
10.  l\_disjoint(\mBbbZ{};L;mFOL-boundvars(fmla'))
11.  mFOL-freevars(mFOquant(isall;z;fmla'))  =  mFOL-freevars(mFOquant(isall;z;body))
\mvdash{}  (FOQuantifier(isall)  z  mFOL-abstract(fmla'))  =  (FOQuantifier(isall)  z  mFOL-abstract(body))


By

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Latex:
(InstLemma  `FOQuantifier\_wf`  [\mkleeneopen{}mFOL-freevars(fmla')\mkleeneclose{};\mkleeneopen{}isall\mkleeneclose{}]\mcdot{}  THEN  Auto)

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