**Description:** Deduction from equality to equivalence of membership. (Contributed by NM, 21-Jun-1993) Allow shortening of eleq1 . (Revised by Wolf
Lammen, 20-Nov-2019)

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

Hypothesis | eleq1d.1 | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) | |

Assertion | eleq1d | ⊢ ( 𝜑 → ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶 ) ) |

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

1 | eleq1d.1 | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) | |

2 | 1 | eqeq2d | ⊢ ( 𝜑 → ( 𝑥 = 𝐴 ↔ 𝑥 = 𝐵 ) ) |

3 | 2 | anbi1d | ⊢ ( 𝜑 → ( ( 𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐶 ) ↔ ( 𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐶 ) ) ) |

4 | 3 | exbidv | ⊢ ( 𝜑 → ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐶 ) ↔ ∃ 𝑥 ( 𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐶 ) ) ) |

5 | dfclel | ⊢ ( 𝐴 ∈ 𝐶 ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐶 ) ) | |

6 | dfclel | ⊢ ( 𝐵 ∈ 𝐶 ↔ ∃ 𝑥 ( 𝑥 = 𝐵 ∧ 𝑥 ∈ 𝐶 ) ) | |

7 | 4 5 6 | 3bitr4g | ⊢ ( 𝜑 → ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶 ) ) |