Proof: Simplify Rearrange Sum 6

Let's prove the following theorem:

if the following are true:
  • x = c + d
  • y = e + f

then ((a + b) + (c + d)) + (e + f) = ((a + b) + x) + y

Proof:

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

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