Invert logical conditions or conditional statements

Question

I want to manually invert a logical condition.

Let's say we have this:

$a = true;
$b = false;

if (!$a or $b) {
    echo 'true';
}

if (!(!$a or $b)) { //I want to solve !(!$a or $b) to a form without ()
    echo 'true 2';
}

What are the rules?

Answer 1

So let say

!(!$a or $b) === true is the same as !$a or $b === false.

So !$a or $b must be false, which means !$a and $b should be both false, which in turn means $a must be true and $b must be false.

In the end, this !(!$a or $b) is equivalent to $a and !$b.

$a  $b  !(!$a or $b)   $a and !$b
T   T   F               F
T   F   T               T
F   T   F               F
F   F   F               F

We can also prove this using Boolean algebra and De Morgan’s theorem

¬(¬A∨B) = ¬(¬A)∧¬B = A∧¬B

Answer 2

According to De Morgan's laws:

!(A and B) == !A or !B
!(A or B)  == !A and !B

Therefore

!(!$a or $b) == !!$a and !$b == $a and !$b

You can also draw up an evaluation table:

$a     $b     !(!$a or $b)  $a and !$b
true   true   false         false
true   false  true          true
false  true   false         false
false  true   false         false

Side note:

Careful with and/or when you have assignments like $val = true and false;, cf. 'AND' vs '&&' as operator