This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 )