**Description:** Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004) (Proof shortened by Andrew Salmon, 26-Jun-2011)

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

Hypotheses | ineq1d.1 | $${\u22a2}{\phi}\to {A}={B}$$ | |

ineq12d.2 | $${\u22a2}{\phi}\to {C}={D}$$ | ||

Assertion | ineq12d | $${\u22a2}{\phi}\to {A}\cap {C}={B}\cap {D}$$ |

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

1 | ineq1d.1 | $${\u22a2}{\phi}\to {A}={B}$$ | |

2 | ineq12d.2 | $${\u22a2}{\phi}\to {C}={D}$$ | |

3 | ineq12 | $${\u22a2}\left({A}={B}\wedge {C}={D}\right)\to {A}\cap {C}={B}\cap {D}$$ | |

4 | 1 2 3 | syl2anc | $${\u22a2}{\phi}\to {A}\cap {C}={B}\cap {D}$$ |