I'm trying to learn JavaScript by going through some code in an application and I keep seeing !variable in if conditions. For example: 
if (!variable.onsubmit || (variable.onsubmit() != false)) {
What is it? Some kind of test if the variable is empty?
! is a logic reversal operator, if something was true it will change it to false, if something is false, it will change to true. example, we know that empty string or 0 in boolean is false.
The exclamation mark (non-null assertion) operator removes null and undefined from the type of an expression. It is used when we we know that a variable that TypeScript thinks could be null or undefined actually isn't.
What is the TypeScript exclamation mark? The non-null assertion operator tells the TypeScript compiler that a value typed as optional cannot be null or undefined . For example, if we define a variable as possibly a string or undefined, the !
In programming and scripting languages, the exclamation mark is used as "not." For example, "! =" also means not equal. See our operator definition for further information. Used to help identify a nonexecutable statement.
! is the logical not operator in JavaScript.
!expression is read as:
expression and evaluate it. In your case that's variable.onsubmit
onsubmit is likely a function, it means - if the function is null or undefined - return false, otherwise return true.In your case !variable.onsubmit means return true if there isn't a function defined (and thus is falsy), otherwise return false (since there is a function defined).
Simply put - !variable means take the truth value of variable and negate it.
Thus, if (!variable) { will enter the if clause if variable is false (or coerces to false)
if (!variable.onsubmit || (variable.onsubmit() != false)) {
Means - check if variable.onsubmit is defined and truthy (thus true), then it checks if calling onsubmit returns a result that coerces to true. In a short line it checks if there is no onsubmit or it returns true.
It is a negation operator used for truth tests on a variable.
var myVariable = 1;
if ( ! myVariable )
{
    // myVariable evaluates as false
}
if ( myVariable )
{
    // myVariable evaluates as true
}
                        The selected answer already answers the question. One thing to add in this is that ! operator can be used in a rather interesting fashion.
obj = {}
if (!!obj) {console.log('works')} // !!obj = true
obj = undefined
if (!!obj) {console.log('does not work')} // !!obj = false
So double !! will change any expression to a boolean value with no exceptions.  
This has a very peculiar use case in some cases.
Lets say there is a function that returns either true or false and nothing else. In that case one could just change return returnValue to return !!returnValue for extra caution (although not need in most of the cases)
Conversely, if a function only accepts true or false and nothing else, one could just call the function as functionA(!!parameter) instead of functionA(parameter), which again is not needed in most of the cases, but could ensure extra security 
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