Provable formulas

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Distributivities

$A\plus (B\with C) \limp (A\plus B)\with (A\plus C)$

$A\tens (B\parr C) \limp (A\tens B)\parr C$

Factorizations

$(A\with B)\plus (A\with C) \limp A\with (B\plus C)$

In many of the above cases the converse implication does not hold.

Exponentials

Provable formulas involving exponential connectives only provide us with the lattice of exponential modalities.