| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | idn2 44633 | . . . . . . 7
⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   ) | 
| 2 |  | 3mix3 1333 | . . . . . . . . . 10
⊢ (𝑥 = 𝐴 → (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)) | 
| 3 | 1, 2 | e2 44651 | . . . . . . . . 9
⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)   ) | 
| 4 |  | abid 2718 | . . . . . . . . 9
⊢ (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)} ↔ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)) | 
| 5 | 3, 4 | e2bir 44653 | . . . . . . . 8
⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)}   ) | 
| 6 |  | dftp2 4691 | . . . . . . . . 9
⊢ {𝐶, 𝐷, 𝐴} = {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)} | 
| 7 | 6 | eleq2i 2833 | . . . . . . . 8
⊢ (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)}) | 
| 8 | 5, 7 | e2bir 44653 | . . . . . . 7
⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝑥 ∈ {𝐶, 𝐷, 𝐴}   ) | 
| 9 |  | eleq1 2829 | . . . . . . . 8
⊢ (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝐴 ∈ {𝐶, 𝐷, 𝐴})) | 
| 10 | 9 | biimpd 229 | . . . . . . 7
⊢ (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} → 𝐴 ∈ {𝐶, 𝐷, 𝐴})) | 
| 11 | 1, 8, 10 | e22 44691 | . . . . . 6
⊢ (   𝐴 ∈ 𝐵   ,   𝑥 = 𝐴   ▶   𝐴 ∈ {𝐶, 𝐷, 𝐴}   ) | 
| 12 | 11 | in2 44625 | . . . . 5
⊢ (   𝐴 ∈ 𝐵   ▶   (𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})   ) | 
| 13 | 12 | gen11 44636 | . . . 4
⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑥(𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})   ) | 
| 14 |  | 19.23v 1942 | . . . 4
⊢
(∀𝑥(𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) ↔ (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})) | 
| 15 | 13, 14 | e1bi 44649 | . . 3
⊢ (   𝐴 ∈ 𝐵   ▶   (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})   ) | 
| 16 |  | idn1 44594 | . . . 4
⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   ) | 
| 17 |  | elisset 2823 | . . . 4
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | 
| 18 | 16, 17 | e1a 44647 | . . 3
⊢ (   𝐴 ∈ 𝐵   ▶   ∃𝑥 𝑥 = 𝐴   ) | 
| 19 |  | id 22 | . . 3
⊢
((∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) → (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})) | 
| 20 | 15, 18, 19 | e11 44708 | . 2
⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ {𝐶, 𝐷, 𝐴}   ) | 
| 21 | 20 | in1 44591 | 1
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) |