Proof of Theorem prtlem17
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | df-rex 3071 | . . 3
⊢
(∃𝑦 ∈
𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦))) | 
| 2 |  | an32 646 | . . . . . . . 8
⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝑥) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) ∧ 𝑦 ∈ 𝐴)) | 
| 3 |  | prtlem14 38875 | . . . . . . . . . . 11
⊢ (Prt
𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))) | 
| 4 |  | elequ2 2123 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦)) | 
| 5 | 4 | biimprd 248 | . . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑥)) | 
| 6 | 3, 5 | syl8 76 | . . . . . . . . . 10
⊢ (Prt
𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ∈ 𝑦) → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑥)))) | 
| 7 | 6 | exp4a 431 | . . . . . . . . 9
⊢ (Prt
𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑧 ∈ 𝑥 → (𝑧 ∈ 𝑦 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑥))))) | 
| 8 | 7 | impd 410 | . . . . . . . 8
⊢ (Prt
𝐴 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝑥) → (𝑧 ∈ 𝑦 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑥)))) | 
| 9 | 2, 8 | biimtrrid 243 | . . . . . . 7
⊢ (Prt
𝐴 → (((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) ∧ 𝑦 ∈ 𝐴) → (𝑧 ∈ 𝑦 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑥)))) | 
| 10 | 9 | expd 415 | . . . . . 6
⊢ (Prt
𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑦 ∈ 𝐴 → (𝑧 ∈ 𝑦 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑥))))) | 
| 11 | 10 | imp5a 440 | . . . . 5
⊢ (Prt
𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑦 ∈ 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝑥)))) | 
| 12 | 11 | imp4b 421 | . . . 4
⊢ ((Prt
𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥)) → ((𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦)) → 𝑤 ∈ 𝑥)) | 
| 13 | 12 | exlimdv 1933 | . . 3
⊢ ((Prt
𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥)) → (∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦)) → 𝑤 ∈ 𝑥)) | 
| 14 | 1, 13 | biimtrid 242 | . 2
⊢ ((Prt
𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥)) → (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝑥)) | 
| 15 | 14 | ex 412 | 1
⊢ (Prt
𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑤 ∈ 𝑦) → 𝑤 ∈ 𝑥))) |