| Step | Hyp | Ref
| Expression |
| 1 | | df-rab 3437 |
. . . 4
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
| 2 | 1 | neeq1i 3005 |
. . 3
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ≠ ∅) |
| 3 | | rabn0 4389 |
. . 3
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝜑) |
| 4 | | n0 4353 |
. . 3
⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ≠ ∅ ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
| 5 | 2, 3, 4 | 3bitr3i 301 |
. 2
⊢
(∃𝑥 ∈
𝐴 𝜑 ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
| 6 | | vex 3484 |
. . . . . 6
⊢ 𝑧 ∈ V |
| 7 | 6 | snss 4785 |
. . . . 5
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ {𝑧} ⊆ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
| 8 | | ssab2 4079 |
. . . . . 6
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 |
| 9 | | sstr2 3990 |
. . . . . 6
⊢ ({𝑧} ⊆ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 → {𝑧} ⊆ 𝐴)) |
| 10 | 8, 9 | mpi 20 |
. . . . 5
⊢ ({𝑧} ⊆ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → {𝑧} ⊆ 𝐴) |
| 11 | 7, 10 | sylbi 217 |
. . . 4
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → {𝑧} ⊆ 𝐴) |
| 12 | | simpr 484 |
. . . . . . . 8
⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) → [𝑧 / 𝑥]𝜑) |
| 13 | | equsb1v 2105 |
. . . . . . . . 9
⊢ [𝑧 / 𝑥]𝑥 = 𝑧 |
| 14 | | velsn 4642 |
. . . . . . . . . 10
⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧) |
| 15 | 14 | sbbii 2076 |
. . . . . . . . 9
⊢ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ↔ [𝑧 / 𝑥]𝑥 = 𝑧) |
| 16 | 13, 15 | mpbir 231 |
. . . . . . . 8
⊢ [𝑧 / 𝑥]𝑥 ∈ {𝑧} |
| 17 | 12, 16 | jctil 519 |
. . . . . . 7
⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) → ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑)) |
| 18 | | df-clab 2715 |
. . . . . . . 8
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) |
| 19 | | sban 2080 |
. . . . . . . 8
⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)) |
| 20 | 18, 19 | bitri 275 |
. . . . . . 7
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)) |
| 21 | | df-rab 3437 |
. . . . . . . . 9
⊢ {𝑥 ∈ {𝑧} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)} |
| 22 | 21 | eleq2i 2833 |
. . . . . . . 8
⊢ (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)}) |
| 23 | | df-clab 2715 |
. . . . . . . 8
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑)) |
| 24 | | sban 2080 |
. . . . . . . 8
⊢ ([𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑)) |
| 25 | 22, 23, 24 | 3bitri 297 |
. . . . . . 7
⊢ (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑)) |
| 26 | 17, 20, 25 | 3imtr4i 292 |
. . . . . 6
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → 𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑}) |
| 27 | 26 | ne0d 4342 |
. . . . 5
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → {𝑥 ∈ {𝑧} ∣ 𝜑} ≠ ∅) |
| 28 | | rabn0 4389 |
. . . . 5
⊢ ({𝑥 ∈ {𝑧} ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ {𝑧}𝜑) |
| 29 | 27, 28 | sylib 218 |
. . . 4
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∃𝑥 ∈ {𝑧}𝜑) |
| 30 | | vsnex 5434 |
. . . . 5
⊢ {𝑧} ∈ V |
| 31 | | sseq1 4009 |
. . . . . 6
⊢ (𝑦 = {𝑧} → (𝑦 ⊆ 𝐴 ↔ {𝑧} ⊆ 𝐴)) |
| 32 | | rexeq 3322 |
. . . . . 6
⊢ (𝑦 = {𝑧} → (∃𝑥 ∈ 𝑦 𝜑 ↔ ∃𝑥 ∈ {𝑧}𝜑)) |
| 33 | 31, 32 | anbi12d 632 |
. . . . 5
⊢ (𝑦 = {𝑧} → ((𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑) ↔ ({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑))) |
| 34 | 30, 33 | spcev 3606 |
. . . 4
⊢ (({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑)) |
| 35 | 11, 29, 34 | syl2anc 584 |
. . 3
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑)) |
| 36 | 35 | exlimiv 1930 |
. 2
⊢
(∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑)) |
| 37 | 5, 36 | sylbi 217 |
1
⊢
(∃𝑥 ∈
𝐴 𝜑 → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑)) |