I am currently studying Dirk Van Dalen's Logic and Structure, in which I encountered two very similar concepts:
Semantic consequence:$ φ \vDash θ$ and $ θ \vDash σ$ $$\Rightarrow φ \vDash σ $$
A tautology using the implication connective:$$ \vDash ((φ \rightarrow θ) \land (θ \rightarrow σ)) \rightarrow (φ \rightarrow σ)$$
I have come to understand that $\Rightarrow$ is taken to be a meta-logical symbol,one that the reader uses to reason; And that $\rightarrow$ is taken to be one of the connectives in the set PROP, with which one has constructed the formal language.
My question is: In the everyday reasoning of a mathematician trying to prove something, is he using semantic consequece?
Or is it something only utilized by the logician, trying to construct tautologies for the mathematician to use?