lo.logic - Is there a formula phi s.t. phi and not-phi have a stronger consistency?

Now that I know the answer, I've found my own simple proof. Probably it's interesting to someone else, so I post it:

I want to show that Con(Σ) is equivalent to ( Con(Σ + φ) OR Con(Σ + not φ) )

Proof:
Con(Σ + φ) OR Con(Σ + not φ) iff

( Σ doesn't prove [φ -> FALSE] ) OR
( Σ doesn't prove [not φ -> FALSE] ) iff

Σ doesn't prove [(not φ -> FALSE) AND (φ -> FALSE)] iff

Σ doesn't prove [FALSE], which is Con(Σ).