Rule of Association/Disjunction/Formulation 2/Reverse Implication

Theorem

 * $\vdash \left({p \lor \left({q \lor r}\right)}\right) \impliedby \left({\left({p \lor q}\right) \lor r}\right)$

Proof
By definition of $\impliedby$, we prove:


 * $\vdash \left({\left({p \lor q}\right) \lor r}\right) \implies \left({p \lor \left({q \lor r}\right)}\right)$

{{TableauLine |n = 16 |f = \paren{ \paren{ p \lor \paren{ q \lor r } } \lor \paren{ p \lor q } } \implies \paren{ \paren{ p \lor \paren{ q \lor r } } \lor \paren{ p \lor \paren{ q \lor r } } } |rlnk = Definition:Hilbert Proof System/Instance 2 |rtxt = Rule $RST \, 1$ |dep = 14 |c = $\paren{ p \lor \paren{ q \lor r } \,/\, r, r \,/\, s$ }}