I'm studying rules of inference with this YouTube video by TheTrevTutor. In one proof, I can't understand how $\phi \to \psi$ gets replaced by $\neg\phi \vee \psi$ (Step 6 to Step 7).
What name of this rule? The video calls it "the definition of $\vee$ with respect to the arrow".
If someone would suggest name of this rule I could google it and know more about it.
Thank you.
The formulae $\phi\rightarrow\psi$ (implication) and $\neg\phi\vee\psi$ (not $\phi$ or $\psi$)have the same semantics, i.e., if you assign truth values to $\phi$ and $\psi$, both formulae have the same truth values.
The implication ($\phi \to \psi$) means that one of the following scenarios is occuring: both $\psi$ and $\phi$ are false, both are true, or $\psi$ is true and $\phi$ is false [this only excludes the scenario where $\phi$ is true and $\psi$ is false].
The "or" statement ($\neg\phi \vee \psi$) is only false in the scenario where $\phi$ is true and $\psi$ is false, which is exactly when the implication is false.
