**Description:** Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007)

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

Hypothesis | necon3bbid.1 | $${\u22a2}{\phi}\to \left({\psi}\leftrightarrow {A}={B}\right)$$ | |

Assertion | necon3bbid | $${\u22a2}{\phi}\to \left(\neg {\psi}\leftrightarrow {A}\ne {B}\right)$$ |

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

1 | necon3bbid.1 | $${\u22a2}{\phi}\to \left({\psi}\leftrightarrow {A}={B}\right)$$ | |

2 | 1 | bicomd | $${\u22a2}{\phi}\to \left({A}={B}\leftrightarrow {\psi}\right)$$ |

3 | 2 | necon3abid | $${\u22a2}{\phi}\to \left({A}\ne {B}\leftrightarrow \neg {\psi}\right)$$ |

4 | 3 | bicomd | $${\u22a2}{\phi}\to \left(\neg {\psi}\leftrightarrow {A}\ne {B}\right)$$ |