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Metamath Proof Explorer

Theorem subaddrii

Description: Relationship between subtraction and addition. (Contributed by NM, 16-Dec-2006)

Ref Expression
Hypotheses negidi.1 ⊢ A ∈ ℂ
pncan3i.2 ⊢ B ∈ ℂ
subadd.3 ⊢ C ∈ ℂ
subaddri.4 ⊢ B + C = A
Assertion subaddrii ⊢ A − B = C

Proof

Step Hyp Ref Expression
1 negidi.1 ⊢ A ∈ ℂ
2 pncan3i.2 ⊢ B ∈ ℂ
3 subadd.3 ⊢ C ∈ ℂ
4 subaddri.4 ⊢ B + C = A
5 1 2 3 subaddi ⊢ A − B = C ↔ B + C = A
6 4 5 mpbir ⊢ A − B = C