So I have been proving various logical statements using induction method (like structural induction , strong induction , weak induction etc ).I was wondering If there is a proof of this "Induction proof method" . So far , I came to this ,
Induction $\rightarrow$ Well ordering principal $\rightarrow$ Axiom of choice $\rightarrow$ ZFC $\rightarrow$ First-order logic theory
So now I wonder , Is there a way to prove (or show equivalence of) this method of proof using just Logic and no Set theory.Also point out if there is a flaw in my reasoning
Edit:It seems like structural induction doesn't do induction over numbers of any kind , it does in on structures .So I can't use peanos axioms to formulate it .I need ZFC .But ZFC is just a kind of first order logic.So structural induction comes from this particular first order logic . But there are some General theorems (which probably don't necessarily belong to ZFC) in Propositional Calculus which I have to prove using structural Induction .But Structural Induction can only be used inside ZFC , not outside of it.I am confused.In a simpler way , The following general theorem I will show at the end of my question is outside of set theory . And I need structural induction to prove it . But structural Induction can only prove things inside set theory. Because Structural Induction is a axiom of Axiomatic set theory.
I will give just a example of one of these general theorem.
"Assume $A$$1$ $\equiv$ $A$$2$ . Show that for any formula $C$-containing $A$$1$ as a part , if we replace one of more occurences of the part $A$$1$ by $A$$2$ , then the resulting formula is logically equivalent to $C$."