How to Left-align Text using Math Mode's Alignment Operator?

Alignment markers alternate alignment between left and right, which is why the the part before the first & is right aligned, and the part between the first and second & is left aligned. You can achieve the behavior you want by doubling up to && so that it toggles twice, cancelling each other out, turning right & left & right into right & left && left.

For your case this would be:

#set page(height: auto, width: auto, margin: 20pt)

$
&not exists x exists y (Q(x, y) <--> Q(y, x)) #h(160pt) &&"Original" \
equiv &forall x not exists y (Q(x,y) <--> Q(y, x)) &&"De Morgan's Law" \
equiv &forall x forall y space not (Q(x,y) <--> Q(y, x)) &&"De Morgan's Law" \
equiv &forall x forall y space ((Q(x,y) and not Q(y,x)) or (not Q(x,y) and Q(y, x))) &&"By (1)"
$
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