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Metamath Proof Explorer

Theorem subaddi

Description: Relationship between subtraction and addition. (Contributed by NM, 26-Nov-1994) (Revised by Mario Carneiro, 21-Dec-2013)

Ref Expression
Hypotheses negidi.1 ⊢ A ∈ ℂ
pncan3i.2 ⊢ B ∈ ℂ
subadd.3 ⊢ C ∈ ℂ
Assertion subaddi ⊢ A − B = C ↔ B + C = A

Proof

Step Hyp Ref Expression
1 negidi.1 ⊢ A ∈ ℂ
2 pncan3i.2 ⊢ B ∈ ℂ
3 subadd.3 ⊢ C ∈ ℂ
4 subadd ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A − B = C ↔ B + C = A
5 1 2 3 4 mp3an ⊢ A − B = C ↔ B + C = A