**Description:** Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004)

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

Hypotheses | eleqtrrd.1 | $${\u22a2}{\phi}\to {A}\in {B}$$ | |

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

Assertion | eleqtrrd | $${\u22a2}{\phi}\to {A}\in {C}$$ |

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

1 | eleqtrrd.1 | $${\u22a2}{\phi}\to {A}\in {B}$$ | |

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

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

4 | 1 3 | eleqtrd | $${\u22a2}{\phi}\to {A}\in {C}$$ |