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Metamath Proof Explorer

Theorem neneqd

Description: Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011)

Ref Expression
Hypothesis neneqd.1 ⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
Assertion neneqd ( 𝜑 → ¬ 𝐴 = 𝐵 )

Proof

Step Hyp Ref Expression
1 neneqd.1 ⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
2 df-ne ⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
3 1 2 sylib ⊢ ( 𝜑 → ¬ 𝐴 = 𝐵 )