How to prove this distributive law using natural deduction

Question

$(q \lor r)\wedge p\vdash(q\wedge p)\lor (r\wedge p)$

After making the first assumption and splitting it up using ∧-elimination, I get stuck.

Can anyone help?

Answer 1

In natural deduction, we will have something like the following deduction rule, $\lor$-elimination:

From $\Gamma, \phi \vdash \chi$ and $\Delta, \psi \vdash \chi$, infer $\Gamma, \Delta, \phi \lor \psi \vdash \chi$.

where $\phi, \psi, \chi$ are formulas, and $\Gamma, \Delta$ are sets of formulas. The name stems from the fact that in the last sequent, the antecedent contains a disjunction that the consequent doesn't.

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In the present case, we are seeking to prove: $$q \lor r, p \vdash (q \land p) \lor (r \land p)$$

Can you see how to use $\lor$-elimination to achieve this?