I want to show :
Every formula of propositional logic A except atoms can be written uniquely to one of these formulas $(B\wedge C)$,$(B \vee C)$ ,$(B \rightarrow C)$ or $(\neg B)$ which $B,C$ are propositional logic.
I tried to prove this by induction on the length of A.
Base case : I suppose A has minimum length ,because A is proposition and is not atom so it should be $(p_1 \wedge p_2)$ or $(p_1 \vee p_2)$ or $(p_1 \rightarrow p_2)$ or $(\neg p_1)$ and $p_i$ are atoms but I don't have any idea that A is unique.
Induction hypotheses:
Suppose $A_1,A_2$ are propositional logics that can be written uniquely to $(B_1\wedge C_1)$,$(B_1 \vee C_1)$ ,$(B_1 \rightarrow C_1)$ or $(\neg B)$ .
Now I don't know how to show that if $A = (A_1 \vee A_2 )$,it is unique formula for $A$.
If I don't prove with induction if I suppose that $A = (...(A_1o_1A_2)o_3(A_3o_3A_4)o_4...A_k)...)$ which $o_i \in \{\wedge,\vee,\rightarrow\}$ and $A_i = p_i \;or\; A_i = (\neg p_i)$ for $i = 1 ... k$ and $p_i$ are atoms now I don't know how to continue!!
Can anyone help me to prove this?