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Metamath Proof Explorer

Theorem necon1abii

Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007) (Proof shortened by Wolf Lammen, 25-Nov-2019)

Ref Expression
Hypothesis necon1abii.1 ⊢ ( ¬ 𝜑 ↔ 𝐴 = 𝐵 )
Assertion necon1abii ( 𝐴 ≠ 𝐵 ↔ 𝜑 )

Proof

Step Hyp Ref Expression
1 necon1abii.1 ⊢ ( ¬ 𝜑 ↔ 𝐴 = 𝐵 )
2 notnotb ⊢ ( 𝜑 ↔ ¬ ¬ 𝜑 )
3 1 necon3bbii ⊢ ( ¬ ¬ 𝜑 ↔ 𝐴 ≠ 𝐵 )
4 2 3 bitr2i ⊢ ( 𝐴 ≠ 𝐵 ↔ 𝜑 )