Proof: Algebra Divide

Let's prove the following theorem:

if a / b = c / d, then d / b = c / a

Proof:

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Given
1 a / b = c / d
Proof Table
# Claim Reason
1 (a / b) ⋅ d = (c / d) ⋅ d if a / b = c / d, then (a / b) ⋅ d = (c / d) ⋅ d
2 (c / d) ⋅ d = c (c / d) ⋅ d = c
3 (a / b) ⋅ d = c if (a / b) ⋅ d = (c / d) ⋅ d and (c / d) ⋅ d = c, then (a / b) ⋅ d = c
4 (a / b) ⋅ d = (d / b) ⋅ a (a / b) ⋅ d = (d / b) ⋅ a
5 (d / b) ⋅ a = c if (a / b) ⋅ d = (d / b) ⋅ a and (a / b) ⋅ d = c, then (d / b) ⋅ a = c
6 ((d / b) ⋅ a) / a = c / a if (d / b) ⋅ a = c, then ((d / b) ⋅ a) / a = c / a
7 ((d / b) ⋅ a) / a = d / b ((d / b) ⋅ a) / a = d / b
8 d / b = c / a if ((d / b) ⋅ a) / a = d / b and ((d / b) ⋅ a) / a = c / a, then d / b = c / a

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