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Metamath Proof Explorer

Theorem nnncan2

Description: Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005)

Ref Expression
Assertion nnncan2 ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A - C - B − C = A − B

Proof

Step Hyp Ref Expression
1 subcl ⊢ B ∈ ℂ ∧ C ∈ ℂ → B − C ∈ ℂ
2 1 3adant1 ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → B − C ∈ ℂ
3 sub32 ⊢ A ∈ ℂ ∧ B − C ∈ ℂ ∧ C ∈ ℂ → A - B − C - C = A - C - B − C
4 2 3 syld3an2 ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A - B − C - C = A - C - B − C
5 nnncan ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A - B − C - C = A − B
6 4 5 eqtr3d ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A - C - B − C = A − B