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Metamath Proof Explorer

Theorem subeq0

Description: If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999)

Ref Expression
Assertion subeq0 ⊢ A ∈ ℂ ∧ B ∈ ℂ → A − B = 0 ↔ A = B

Proof

Step Hyp Ref Expression
1 subid ⊢ B ∈ ℂ → B − B = 0
2 1 adantl ⊢ A ∈ ℂ ∧ B ∈ ℂ → B − B = 0
3 2 eqeq2d ⊢ A ∈ ℂ ∧ B ∈ ℂ → A − B = B − B ↔ A − B = 0
4 subcan2 ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ B ∈ ℂ → A − B = B − B ↔ A = B
5 4 3anidm23 ⊢ A ∈ ℂ ∧ B ∈ ℂ → A − B = B − B ↔ A = B
6 3 5 bitr3d ⊢ A ∈ ℂ ∧ B ∈ ℂ → A − B = 0 ↔ A = B