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I am struggling with the way to write a clear and mathematical proof of logical theorems. Take for example the theorem $\Gamma \models A, \Gamma \subseteq \Delta$ implies $\Delta \models A$. I can prove it this way:

If every formula in $\Delta$ is validated, then that validates every formula in $\Gamma$, which validates $A$.

But how do I prove such thing with mathematical language?

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What exactly dies "validated" mean here? Spell that out a bit, and you will have your desired proof. –  Peter Smith May 15 '13 at 18:19
If the word "validated" was used in the definition of $\models$ (and was, therefore, defined beforehand), then you might not need to spell it out, but otherwise I agree with Peter that you should. –  Andreas Blass May 15 '13 at 20:55
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