**Description:** Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007)

Ref | Expression | ||
---|---|---|---|

Hypothesis | necon2bd.1 | $${\u22a2}{\phi}\to \left({\psi}\to {A}\ne {B}\right)$$ | |

Assertion | necon2bd | $${\u22a2}{\phi}\to \left({A}={B}\to \neg {\psi}\right)$$ |

Step | Hyp | Ref | Expression |
---|---|---|---|

1 | necon2bd.1 | $${\u22a2}{\phi}\to \left({\psi}\to {A}\ne {B}\right)$$ | |

2 | df-ne | $${\u22a2}{A}\ne {B}\leftrightarrow \neg {A}={B}$$ | |

3 | 1 2 | syl6ib | $${\u22a2}{\phi}\to \left({\psi}\to \neg {A}={B}\right)$$ |

4 | 3 | con2d | $${\u22a2}{\phi}\to \left({A}={B}\to \neg {\psi}\right)$$ |