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A question in substitution in logic.

  1. Jul 22, 2006 #1
    i need to prove that if x is the first of sub([tex]\phi[/tex];a,[tex]\psi[/tex]) then there exists 1<=i<=n and there exist firsts [tex]\phi' of \phi_i and \psi' for \psi[/tex] such that x=sub([tex]\phi';a,\psi[/tex])[tex]\psi'[/tex]

    where sub(t;a,b) is defined as follows:
    let a1,..,an be n signs and b1,..,bn expressions.
    the substitution sub(t;a,b) is defined as:
    if t is t1,...,tk then sub(t;a,b)=x1x2...xk
    when 1<=i<=k xi is bj if ti=aj and xi is ti when ti isnt in {a1,...,an}.

    what i did is as follows:
    x is the first of sub(t;a,b) then there exists y such that sub(t;a,b)=xy
    and let t' be the first of t, then t=t'z and thuse we can deduce that:
    but i dont know how to procceed from here.
    thanks in davance.
    Last edited: Jul 22, 2006
  2. jcsd
  3. Jul 23, 2006 #2
    i have another question:
    let a|=a' and b|=b' then prove that a'->b|=a->b' without using ~a'|=~a and metrial conditional, i.e without using ~PvQ=P->Q.
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