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 ( 𝜑𝐴𝐵 )

Proof

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