**Description:** A modus tollens deduction for inequality. (Contributed by Steven
Nguyen, 1-Jun-2023)

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

Hypotheses | mteqand.1 | $${\u22a2}{\phi}\to {C}\ne {D}$$ | |

mteqand.2 | $${\u22a2}\left({\phi}\wedge {A}={B}\right)\to {C}={D}$$ | ||

Assertion | mteqand | $${\u22a2}{\phi}\to {A}\ne {B}$$ |

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

1 | mteqand.1 | $${\u22a2}{\phi}\to {C}\ne {D}$$ | |

2 | mteqand.2 | $${\u22a2}\left({\phi}\wedge {A}={B}\right)\to {C}={D}$$ | |

3 | 1 | neneqd | $${\u22a2}{\phi}\to \neg {C}={D}$$ |

4 | 3 2 | mtand | $${\u22a2}{\phi}\to \neg {A}={B}$$ |

5 | 4 | neqned | $${\u22a2}{\phi}\to {A}\ne {B}$$ |