Proof: Substitution in the Middle

Let's prove the following theorem:

if a + b = c, then ((x + a) + b) + y = (x + c) + y

Proof:

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Given
1 a + b = c
Proof Table
# Claim Reason
1 x + (a + b) = x + c if a + b = c, then x + (a + b) = x + c
2 (x + a) + b = x + (a + b) (x + a) + b = x + (a + b)
3 (x + a) + b = x + c if (x + a) + b = x + (a + b) and x + (a + b) = x + c, then (x + a) + b = x + c
4 ((x + a) + b) + y = (x + c) + y if (x + a) + b = x + c, then ((x + a) + b) + y = (x + c) + y

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