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Metamath Proof Explorer

Theorem subsubd

Description: Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses negidd.1 ⊢ φ → A ∈ ℂ
pncand.2 ⊢ φ → B ∈ ℂ
subaddd.3 ⊢ φ → C ∈ ℂ
Assertion subsubd ⊢ φ → A − B − C = A - B + C

Proof

Step Hyp Ref Expression
1 negidd.1 ⊢ φ → A ∈ ℂ
2 pncand.2 ⊢ φ → B ∈ ℂ
3 subaddd.3 ⊢ φ → C ∈ ℂ
4 subsub ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A − B − C = A - B + C
5 1 2 3 4 syl3anc ⊢ φ → A − B − C = A - B + C