**Description:** A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995) (Proof shortened by Andrew Salmon, 25-May-2011)

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

Hypotheses | 3eqtr3d.1 | $${\u22a2}{\phi}\to {A}={B}$$ | |

3eqtr3d.2 | $${\u22a2}{\phi}\to {A}={C}$$ | ||

3eqtr3d.3 | $${\u22a2}{\phi}\to {B}={D}$$ | ||

Assertion | 3eqtr3d | $${\u22a2}{\phi}\to {C}={D}$$ |

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

1 | 3eqtr3d.1 | $${\u22a2}{\phi}\to {A}={B}$$ | |

2 | 3eqtr3d.2 | $${\u22a2}{\phi}\to {A}={C}$$ | |

3 | 3eqtr3d.3 | $${\u22a2}{\phi}\to {B}={D}$$ | |

4 | 1 2 | eqtr3d | $${\u22a2}{\phi}\to {B}={C}$$ |

5 | 4 3 | eqtr3d | $${\u22a2}{\phi}\to {C}={D}$$ |