Satisfiable Set minus Formula is Satisfiable

Theorem

Let $\LL$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\LL$.

Let $\FF$ be an $\mathscr M$-satisfiable set of formulas from $\LL$.

Let $\phi \in \FF$.


Then $\FF \setminus \set \phi$ is also $\mathscr M$-satisfiable.


Proof

This is an immediate consequence of Subset of Satisfiable Set is Satisfiable.

$\blacksquare$


Also see


Sources

  • 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.5.2$: Theorem $2.44$
  • 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.10$: Exercise $2.15$
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