I feel this is a basic question but it has been bugging me.
Suppose I have the following derivation (in intuitionistic logic):
$$(\wedge^{+})\frac{\Gamma_1 \vdash x=1 \hspace{1cm} \Gamma_2 \vdash x=2}{\Gamma_1 \cup \Gamma_2 \vdash x=1 \wedge x=2}$$
Clearly it's not true that $x=1$ and $x=2$ so the sequent $\Gamma_1 \cup \Gamma_2 \vdash x=1 \wedge x=2$ should not be derivable but I cannot see the rule/axiom which forbids this.
Can someone please explain where exactly this derivation is invalid.
It is valid.
The points is though that if $\Gamma_1\vdash x=1$ and $\Gamma_2\vdash x=2$ then $\Gamma_1 \cup \Gamma_2$ is inconsistent. However from an inconsistent set we may derive any conclusion, even in intutionistic logic.