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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 )