Why is $p\Rightarrow q$ equivalent to $\neg p\lor q$ and how to prove it I don't know how to prove that $p\Rightarrow q$ is equivalent to $\neg p\lor q$ ,here is the link p=>q . And I don't know how wolframalpha generate "Minimal forms" . 
Can you prove $p\Rightarrow q \equiv \neg p\lor q$,  and explain how to get "Minimal forms" ?
Thanks!
 A: By definition, $p\Rightarrow q$ is true if and only if the consequence $q$ is true, or the antecedent $p$ is false. You can see it in the truth table that defines the implication. That is, $p\Rightarrow q$ is true if and only if either $\neg p$ is true, or $q$ is true; i.e., if and only if $\neg p\lor q$ is true (what you write as $!p||q$).
Or you can simply look at the truth tables. The truth table of $\neg p\lor q$ is the same as the truth table of $p\Rightarrow q$: true if $p$ and $q$ are false; true if $p$ is false and $q$ is true; false if $p$ is true and $q$ is false; true if $p$ and $q$ are both true:
$$\begin{array}{c|c||c}
p & q & p\Rightarrow q\\
\hline
0 & 0 & 1\\
0 & 1 & 1\\
1 & 0 & 0\\
1 & 1 & 1
\end{array}\qquad\qquad
\begin{array}{c|c|c|c}
p & q & \neg p & \neg p\lor q\\
\hline
0 & 0 & 1 & 1\\
0 & 1 & 1 & 1\\
1 & 0 & 0 & 0\\
1 & 1 & 0 & 1
\end{array}.$$
The final columns are identical, so the two formulas take the same truth values given the same truth inputs: that is, they are propositionally equivalent.
A: Using truth tables is a simple way to prove it.
A: I don't reckon that you need a Truth Table. What do you reckon of the intuitive explanation beneath? 
From: Philip Johnson-Laird BA PhD Psychology (UCL), Stuart Professor of Psychology Emeritus at Princeton.    (Author isn't  a logician.)    How We Reason  (1st edn 2008). p. 108. 
I changed the author's choice of first names, to ones that start with P and Q to fit the title. I symbolized the disjunctions in square brackets.

An exclusive
  disjunction, such as:

Either Pia helped or Quinn helped, but not both 

is equivalent to the proposition:    

Pia  helped or the Quinn helped, and not both Pia helped and the  Quinn 
    helped.

Hence, exclusive disjunction also has a logical meaning.
  In an analogous way we can define a logical meaning of “if”. The sentence:

If Pia  didn’t help then  Quinn  did. [If ¬P, then Q.]

means:

Pia  helped or  Quinn  did, or both. [P ∨ Q]

In its logical meaning, the conditional is compatible with three possibilities: Pia  didn’t help and Quinn did [¬P ∧ Q] , Pia  helped and Quinn didn’t [P ∧ ¬Q], Pia helped and Quinn helped [P ∧ Q]. The only possibility that the conditional
  rules out is that neither the Pia nor Quinn helped [¬P ∧ ¬Q]. The three possibilities
  that the conditional allows are the same as those for the inclusive disjunction.

