Logical Consequence with Union
Surely proving $P$ with or without assuming it are not equivalent situations. The LHS of $\models$ should intuitively be "conjuncted" into one big premise.
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Theorem
Let $U$ be a set of propositional formulas.
Let $P$ be a propositional formula.
Let $U \models P$ denote that $P$ is a semantic consequence of $U$.
Then:
- $U \models P$
iff:
- $U \cup P \models P$
Proof
First we note that, from the Law of Identity:
- $P \models P$
Suppose $U \models P$.
So we have:
- $U \models P$
- $P \models P$
So by definition of model:
- $U \cup P \models P$
It also follows that if $U \cup P \models P$, then:
- $U \models P$
- $P \models P$
Thus we have shown that $U \models P$ iff $U \cup P \models P$.
$\blacksquare$