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Metamath Proof Explorer

Theorem neg2sub

Description: Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007)

Ref Expression
Assertion neg2sub ⊢ A ∈ ℂ ∧ B ∈ ℂ → - A - − B = B − A

Proof

Step Hyp Ref Expression
1 negcl ⊢ A ∈ ℂ → − A ∈ ℂ
2 subneg ⊢ − A ∈ ℂ ∧ B ∈ ℂ → - A - − B = - A + B
3 1 2 sylan ⊢ A ∈ ℂ ∧ B ∈ ℂ → - A - − B = - A + B
4 negsubdi ⊢ A ∈ ℂ ∧ B ∈ ℂ → − A − B = - A + B
5 negsubdi2 ⊢ A ∈ ℂ ∧ B ∈ ℂ → − A − B = B − A
6 3 4 5 3eqtr2d ⊢ A ∈ ℂ ∧ B ∈ ℂ → - A - − B = B − A