# Substitution for propositional logic

In the propositional logic, let the string $$(a_1, a_2, \cdots, a_n)$$ be WFF. And exist natural numbers $$i< j\in \{1,2 \cdots, n\}$$ search that string $$(a_i,a_{i+1}, \cdots , a_j )$$ is also WFF. Prove that if $$(b_1, b_2, \cdots, b_m)$$ is WFF, then $$(a_1, a_2, a_{i-1}, b_1, b_2, \cdots, b_m, a_{j+1}\cdots, a_n)$$ is also WFF.

In my logical book, substitution is defined only for proposition letter. But I think this is true. How can we prove this?

• You are asked to subst into a formula a sub-formula with another formula. – Mauro ALLEGRANZA May 9 at 11:20
• @MauroALLEGRANZA Yes! Thanks! Do you know the proof or material with this topic? – amoogae May 9 at 11:49
• Do you have a link to a link to a copy of the book or the chapter where this exercise appears? – prime4567 May 9 at 11:50
• @prime4567 Sorry. I made this problem. And the book that I read is written in my native.. – amoogae May 9 at 12:16
• IMO the proof is by induction on the compelxity of the formula, considering the specifications for WFF. For simplicity, consider only the connectives : $\lnot, \lor$. – Mauro ALLEGRANZA May 9 at 12:37