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Metamath Proof Explorer

Theorem subnegd

Description: Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses negidd.1 ⊢ φ → A ∈ ℂ
pncand.2 ⊢ φ → B ∈ ℂ
Assertion subnegd ⊢ φ → A − − B = A + B

Proof

Step Hyp Ref Expression
1 negidd.1 ⊢ φ → A ∈ ℂ
2 pncand.2 ⊢ φ → B ∈ ℂ
3 subneg ⊢ A ∈ ℂ ∧ B ∈ ℂ → A − − B = A + B
4 1 2 3 syl2anc ⊢ φ → A − − B = A + B