This video is a straightforward presentation on the material equivalence operator and the rules of replacement. Two statements are materially equivalent when they have the same truth value. The rules of replacement cover various complex statement forms that have the same truth value as each other. Where one appears in an argument, it may be replaced with a materially equivalent form.

I made this video from a webpage I created for the purpose. What follows is the HTML from that page minus the CSS, graphics, and title.

Material Equivalence operator

Material Equivalence:

(P ≡ Q) ≡ [(P ⊃ Q) & (Q ⊃ P)]

(P ≡ Q) ≡ [(P & Q) v (~P & ~Q)]

Double Negation:

P ≡ ~~P

Tautology:

P ≡ (P & P)

P ≡ (P v P)

Commutation:

(P v Q) ≡ (Q v P)

(P & Q) ≡ (Q & P)

(x + y) = (y + x)

(x * y) = (y * x)

Association:

[P v (Q v R)] ≡ [(P v Q) v R]

[P & (Q & R)] ≡ [(P & Q) & R]

(x + (y + z)) = ((x + y) + z)

(x * (y * z)) = ((x * y) * z)

Distribution:

[P & (Q v R)] ≡ [(P & Q) v (P & R)]

[P v (Q & R)] ≡ [(P v Q) & (P v R)]

De Morgan’s Theorems:

~(P & Q) ≡ (~P v ~Q)

~(P v Q) ≡ (~P & ~Q)

!($p && $q) == (!$p || !$q)

!($p || $q) == (!$p && !$q)

Material Implication:

(P ⊃ Q) ≡ (~P v Q)

if p() {
    q();
}

functions the same as

!p() || q();

Transposition:

(P ⊃ Q) ≡ (~Q ⊃ ~P)

P ⊃ Q
~Q
So, ~P

Exportation:

[(P & Q) ⊃ R] ≡ [P ⊃ (Q ⊃ R)]

if ($p && $q) {
    run_code();
}

is logically equivalent to

if ($p) {
    if ($q) {
        run_code();
    }
}