**Description:** Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008)

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

Hypothesis | necon2d.1 | $${\u22a2}{\phi}\to \left({A}={B}\to {C}\ne {D}\right)$$ | |

Assertion | necon2d | $${\u22a2}{\phi}\to \left({C}={D}\to {A}\ne {B}\right)$$ |

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

1 | necon2d.1 | $${\u22a2}{\phi}\to \left({A}={B}\to {C}\ne {D}\right)$$ | |

2 | df-ne | $${\u22a2}{C}\ne {D}\leftrightarrow \neg {C}={D}$$ | |

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

4 | 3 | necon2ad | $${\u22a2}{\phi}\to \left({C}={D}\to {A}\ne {B}\right)$$ |