# What does the disjunction elimination rule say?

The first version (http://www.fecundity.com/logic/) says that:

• if $\Sigma\vdash\phi_0\lor\phi_1$ and $\Sigma\vdash\lnot\phi_0$, then $\Sigma\vdash\phi_1$
• if $\Sigma\vdash\phi_0\lor\phi_1$ and $\Sigma\vdash\lnot\phi_1$, then $\Sigma\vdash\phi_0$

The second version (S. Hedman - A First Course in Logic) says that:

• if $\Sigma\models\phi_0\lor\phi_1$, $\Sigma\cup\left\{\phi_0\right\}\vdash\phi_2$ and $\Sigma\cup\left\{\phi_1\right\}\vdash\phi_2$, then $\Sigma\vdash\phi_2$

Using the first version of the rule, I can't even demonstrate that if $\Sigma\vdash\phi\lor\phi$ then $\Sigma\vdash\phi$. Perhaps the entire system presented by the first source is not complete, in the sense that you can't prove certain true statements. Of course, it may be my fault, instead.

Thanks.

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$$\frac{\Gamma \vdash A \vee B \qquad \Gamma, A\vdash C \qquad \Gamma, B\vdash C}{\Gamma \vdash C}$$