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Full Idea
The style of proof called 'induction on formula construction' (or 'on the number of connectives', or 'on the length of the formula') rest on the fact that all formulas are built up from atomic formulas according to strict rules.
Gist of Idea
Proof by induction 'on the length of the formula' deconstructs a formula into its accepted atoms
Source
Theodore Sider (Logic for Philosophy [2010], 2.7)
Book Ref
Sider,Theodore: 'Logic for Philosophy' [OUP 2010], p.52
A Reaction
Hence the proof deconstructs the formula, and takes it back to a set of atomic formulas have already been established.