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Metamath Proof Explorer

Theorem subeq0i

Description: If the difference between two numbers is zero, they are equal. (Contributed by NM, 8-May-1999)

Ref Expression
Hypotheses negidi.1 ⊢ A ∈ ℂ
pncan3i.2 ⊢ B ∈ ℂ
Assertion subeq0i ⊢ A − B = 0 ↔ A = B

Proof

Step Hyp Ref Expression
1 negidi.1 ⊢ A ∈ ℂ
2 pncan3i.2 ⊢ B ∈ ℂ
3 subeq0 ⊢ A ∈ ℂ ∧ B ∈ ℂ → A − B = 0 ↔ A = B
4 1 2 3 mp2an ⊢ A − B = 0 ↔ A = B