Commutative Property Variation 1
Substitution 2
Subtraction Example
Transitive Property of Equality Variation 2
Transitive Property Application 2
Subtract Substitute 2 Vars

Proof: Subtract Substitute 2 Vars

Let's prove the following theorem:

if the following are true:
  • a = b
  • c = d

then a - c = b - d

Proof:

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