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Step * 1 1 1 1 of Lemma assert-PZF_safe


1. C : Type
2. left : Form(C)
3. right : Form(C)
4. ∀vs:Atom List. (↑(FormSafe2(left) vs) ⇐⇒ FormSafe1''(left) vs)
5. ∀vs:Atom List. (↑(FormSafe2(right) vs) ⇐⇒ FormSafe1''(right) vs)
6. vs : Atom List
7. ¬(vs = [] ∈ (Atom List))
8. x : Atom
9. hd(vs) = x ∈ Atom
10. ∀x:Atom. ((x ∈ vs) ⇒ (vs-[x] = [] ∈ (Atom List) ⇐⇒ set-equal(Atom;vs;[x])))
11. ↑(null(vs-[x])
∧b ((FormVar?(left) ∧b FormVar-name(left) =a x ∧b (¬bx ∈b FormFvs(right)))
   ∨b(FormVar?(right) ∧b FormVar-name(right) =a x ∧b (¬bx ∈b FormFvs(left)))))
12. vs-[x] = [] ∈ (Atom List) ⇐⇒ set-equal(Atom;vs;[x])
⊢ (vs-[x] = [] ∈ (Atom List))
∧ (((↑FormVar?(left)) ∧ (FormVar-name(left) = x ∈ Atom) ∧ (¬(x ∈ FormFvs(right))))
  ∨ ((↑FormVar?(right)) ∧ (FormVar-name(right) = x ∈ Atom) ∧ (¬(x ∈ FormFvs(left)))))
BY
{ (RW assert_pushdownC (-2) THEN Auto) }

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Latex:

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Latex:

1.  C  :  Type
2.  left  :  Form(C)
3.  right  :  Form(C)
4.  \mforall{}vs:Atom  List.  (\muparrow{}(FormSafe2(left)  vs)  \mLeftarrow{}{}\mRightarrow{}  FormSafe1''(left)  vs)
5.  \mforall{}vs:Atom  List.  (\muparrow{}(FormSafe2(right)  vs)  \mLeftarrow{}{}\mRightarrow{}  FormSafe1''(right)  vs)
6.  vs  :  Atom  List
7.  \mneg{}(vs  =  [])
8.  x  :  Atom
9.  hd(vs)  =  x
10.  \mforall{}x:Atom.  ((x  \mmember{}  vs)  {}\mRightarrow{}  (vs-[x]  =  []  \mLeftarrow{}{}\mRightarrow{}  set-equal(Atom;vs;[x])))
11.  \muparrow{}(null(vs-[x])
\mwedge{}\msubb{}  ((FormVar?(left)  \mwedge{}\msubb{}  FormVar-name(left)  =a  x  \mwedge{}\msubb{}  (\mneg{}\msubb{}x  \mmember{}\msubb{}  FormFvs(right)))
      \mvee{}\msubb{}(FormVar?(right)  \mwedge{}\msubb{}  FormVar-name(right)  =a  x  \mwedge{}\msubb{}  (\mneg{}\msubb{}x  \mmember{}\msubb{}  FormFvs(left)))))
12.  vs-[x]  =  []  \mLeftarrow{}{}\mRightarrow{}  set-equal(Atom;vs;[x])
\mvdash{}  (vs-[x]  =  [])
\mwedge{}  (((\muparrow{}FormVar?(left))  \mwedge{}  (FormVar-name(left)  =  x)  \mwedge{}  (\mneg{}(x  \mmember{}  FormFvs(right))))
    \mvee{}  ((\muparrow{}FormVar?(right))  \mwedge{}  (FormVar-name(right)  =  x)  \mwedge{}  (\mneg{}(x  \mmember{}  FormFvs(left)))))


By

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Latex:
(RW  assert\_pushdownC  (-2)  THEN  Auto)

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