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

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

Hypotheses | eqnetrrd.1 | $${\u22a2}{\phi}\to {A}={B}$$ | |

eqnetrrd.2 | $${\u22a2}{\phi}\to {A}\ne {C}$$ | ||

Assertion | eqnetrrd | $${\u22a2}{\phi}\to {B}\ne {C}$$ |

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

1 | eqnetrrd.1 | $${\u22a2}{\phi}\to {A}={B}$$ | |

2 | eqnetrrd.2 | $${\u22a2}{\phi}\to {A}\ne {C}$$ | |

3 | 1 | eqcomd | $${\u22a2}{\phi}\to {B}={A}$$ |

4 | 3 2 | eqnetrd | $${\u22a2}{\phi}\to {B}\ne {C}$$ |