**Description:** Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012)

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

Hypotheses | neeqtrrd.1 | $${\u22a2}{\phi}\to {A}\ne {B}$$ | |

neeqtrrd.2 | $${\u22a2}{\phi}\to {C}={B}$$ | ||

Assertion | neeqtrrd | $${\u22a2}{\phi}\to {A}\ne {C}$$ |

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

1 | neeqtrrd.1 | $${\u22a2}{\phi}\to {A}\ne {B}$$ | |

2 | neeqtrrd.2 | $${\u22a2}{\phi}\to {C}={B}$$ | |

3 | 2 | eqcomd | $${\u22a2}{\phi}\to {B}={C}$$ |

4 | 1 3 | neeqtrd | $${\u22a2}{\phi}\to {A}\ne {C}$$ |