Metamath Proof Explorer


Theorem necon1ai

Description: Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007) (Proof shortened by Wolf Lammen, 22-Nov-2019)

Ref Expression
Hypothesis necon1ai.1 ( ¬ 𝜑𝐴 = 𝐵 )
Assertion necon1ai ( 𝐴𝐵𝜑 )

Proof

Step Hyp Ref Expression
1 necon1ai.1 ( ¬ 𝜑𝐴 = 𝐵 )
2 1 necon3ai ( 𝐴𝐵 → ¬ ¬ 𝜑 )
3 2 notnotrd ( 𝐴𝐵𝜑 )