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Metamath Proof Explorer

Theorem necon2abii

Description: Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007)

Ref Expression
Hypothesis necon2abii.1 ⊢ ( 𝐴 = 𝐵 ↔ ¬ 𝜑 )
Assertion necon2abii ( 𝜑 ↔ 𝐴 ≠ 𝐵 )

Proof

Step Hyp Ref Expression
1 necon2abii.1 ⊢ ( 𝐴 = 𝐵 ↔ ¬ 𝜑 )
2 1 bicomi ⊢ ( ¬ 𝜑 ↔ 𝐴 = 𝐵 )
3 2 necon1abii ⊢ ( 𝐴 ≠ 𝐵 ↔ 𝜑 )
4 3 bicomi ⊢ ( 𝜑 ↔ 𝐴 ≠ 𝐵 )