Proof of Theorem reu8
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | rmo4.1 | . . 3
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | 
| 2 | 1 | cbvreuvw 3403 | . 2
⊢
(∃!𝑥 ∈
𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) | 
| 3 |  | reu6 3731 | . 2
⊢
(∃!𝑦 ∈
𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥)) | 
| 4 |  | dfbi2 474 | . . . . 5
⊢ ((𝜓 ↔ 𝑦 = 𝑥) ↔ ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓))) | 
| 5 | 4 | ralbii 3092 | . . . 4
⊢
(∀𝑦 ∈
𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ ∀𝑦 ∈ 𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓))) | 
| 6 |  | r19.26 3110 | . . . . 5
⊢
(∀𝑦 ∈
𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓))) | 
| 7 |  | ancom 460 | . . . . . 6
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ∧ 𝜑)) | 
| 8 |  | equcom 2016 | . . . . . . . . . 10
⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | 
| 9 | 8 | imbi2i 336 | . . . . . . . . 9
⊢ ((𝜓 → 𝑥 = 𝑦) ↔ (𝜓 → 𝑦 = 𝑥)) | 
| 10 | 9 | ralbii 3092 | . . . . . . . 8
⊢
(∀𝑦 ∈
𝐴 (𝜓 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥)) | 
| 11 | 10 | a1i 11 | . . . . . . 7
⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥))) | 
| 12 |  | biimt 360 | . . . . . . . 8
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 → 𝜑))) | 
| 13 |  | df-ral 3061 | . . . . . . . . 9
⊢
(∀𝑦 ∈
𝐴 (𝑦 = 𝑥 → 𝜓) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓))) | 
| 14 |  | bi2.04 387 | . . . . . . . . . 10
⊢ ((𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓)) ↔ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓))) | 
| 15 | 14 | albii 1818 | . . . . . . . . 9
⊢
(∀𝑦(𝑦 ∈ 𝐴 → (𝑦 = 𝑥 → 𝜓)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓))) | 
| 16 |  | eleq1w 2823 | . . . . . . . . . . . . 13
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | 
| 17 | 16, 1 | imbi12d 344 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜓))) | 
| 18 | 17 | bicomd 223 | . . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → ((𝑦 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜑))) | 
| 19 | 18 | equcoms 2018 | . . . . . . . . . 10
⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜑))) | 
| 20 | 19 | equsalvw 2002 | . . . . . . . . 9
⊢
(∀𝑦(𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜓)) ↔ (𝑥 ∈ 𝐴 → 𝜑)) | 
| 21 | 13, 15, 20 | 3bitrri 298 | . . . . . . . 8
⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)) | 
| 22 | 12, 21 | bitrdi 287 | . . . . . . 7
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓))) | 
| 23 | 11, 22 | anbi12d 632 | . . . . . 6
⊢ (𝑥 ∈ 𝐴 → ((∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ∧ 𝜑) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)))) | 
| 24 | 7, 23 | bitrid 283 | . . . . 5
⊢ (𝑥 ∈ 𝐴 → ((𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (∀𝑦 ∈ 𝐴 (𝜓 → 𝑦 = 𝑥) ∧ ∀𝑦 ∈ 𝐴 (𝑦 = 𝑥 → 𝜓)))) | 
| 25 | 6, 24 | bitr4id 290 | . . . 4
⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ((𝜓 → 𝑦 = 𝑥) ∧ (𝑦 = 𝑥 → 𝜓)) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))) | 
| 26 | 5, 25 | bitrid 283 | . . 3
⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))) | 
| 27 | 26 | rexbiia 3091 | . 2
⊢
(∃𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐴 (𝜓 ↔ 𝑦 = 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) | 
| 28 | 2, 3, 27 | 3bitri 297 | 1
⊢
(∃!𝑥 ∈
𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |