# A question in substitution in logic.

1. Jul 22, 2006

### MathematicalPhysicist

i need to prove that if x is the first of sub($$\phi$$;a,$$\psi$$) then there exists 1<=i<=n and there exist firsts $$\phi' of \phi_i and \psi' for \psi$$ such that x=sub($$\phi';a,\psi$$)$$\psi'$$

where sub(t;a,b) is defined as follows:
let a1,..,an be n signs and b1,..,bn expressions.
a=(a1,...,an)
b=(b1,..,bn)
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:
sub(t;a,b)=sub(t';a,b)sub(z;a,b)=xy
but i dont know how to procceed from here.
thanks in davance.

Last edited: Jul 22, 2006
2. Jul 23, 2006

### MathematicalPhysicist

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.