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Metamath Proof Explorer

Theorem neg11d

Description: If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses negidd.1 ⊢ φ → A ∈ ℂ
pncand.2 ⊢ φ → B ∈ ℂ
neg11d.3 ⊢ φ → − A = − B
Assertion neg11d ⊢ φ → A = B

Proof

Step Hyp Ref Expression
1 negidd.1 ⊢ φ → A ∈ ℂ
2 pncand.2 ⊢ φ → B ∈ ℂ
3 neg11d.3 ⊢ φ → − A = − B
4 1 2 neg11ad ⊢ φ → − A = − B ↔ A = B
5 3 4 mpbid ⊢ φ → A = B