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Metamath Proof Explorer

Theorem subeq0d

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 )
subeq0d.3 ⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = 0 )
Assertion subeq0d ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step Hyp Ref Expression
1 negidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 pncand.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 subeq0d.3 ⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = 0 )
4 subeq0 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )
5 1 2 4 syl2anc ⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )
6 3 5 mpbid ⊢ ( 𝜑 → 𝐴 = 𝐵 )