Step | Hyp | Ref
| Expression |
1 | | df-rab 2921 |
. . . 4
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
2 | 1 | neeq1i 2858 |
. . 3
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ≠ ∅) |
3 | | rabn0 3958 |
. . 3
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝜑) |
4 | | n0 3931 |
. . 3
⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ≠ ∅ ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
5 | 2, 3, 4 | 3bitr3i 290 |
. 2
⊢
(∃𝑥 ∈
𝐴 𝜑 ↔ ∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
6 | | vex 3203 |
. . . . . 6
⊢ 𝑧 ∈ V |
7 | 6 | snss 4316 |
. . . . 5
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ {𝑧} ⊆ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
8 | | ssab2 3686 |
. . . . . 6
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 |
9 | | sstr2 3610 |
. . . . . 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 207 |
. . . 4
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → {𝑧} ⊆ 𝐴) |
12 | | simpr 477 |
. . . . . . . 8
⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) → [𝑧 / 𝑥]𝜑) |
13 | | equsb1 2368 |
. . . . . . . . 9
⊢ [𝑧 / 𝑥]𝑥 = 𝑧 |
14 | | velsn 4193 |
. . . . . . . . . 10
⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧) |
15 | 14 | sbbii 1887 |
. . . . . . . . 9
⊢ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ↔ [𝑧 / 𝑥]𝑥 = 𝑧) |
16 | 13, 15 | mpbir 221 |
. . . . . . . 8
⊢ [𝑧 / 𝑥]𝑥 ∈ {𝑧} |
17 | 12, 16 | jctil 560 |
. . . . . . 7
⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) → ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑)) |
18 | | df-clab 2609 |
. . . . . . . 8
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) |
19 | | sban 2399 |
. . . . . . . 8
⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)) |
20 | 18, 19 | bitri 264 |
. . . . . . 7
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)) |
21 | | df-rab 2921 |
. . . . . . . . 9
⊢ {𝑥 ∈ {𝑧} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)} |
22 | 21 | eleq2i 2693 |
. . . . . . . 8
⊢ (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)}) |
23 | | df-clab 2609 |
. . . . . . . . 9
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑)) |
24 | | sban 2399 |
. . . . . . . . 9
⊢ ([𝑧 / 𝑥](𝑥 ∈ {𝑧} ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑)) |
25 | 23, 24 | bitri 264 |
. . . . . . . 8
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ {𝑧} ∧ 𝜑)} ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑)) |
26 | 22, 25 | bitri 264 |
. . . . . . 7
⊢ (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} ↔ ([𝑧 / 𝑥]𝑥 ∈ {𝑧} ∧ [𝑧 / 𝑥]𝜑)) |
27 | 17, 20, 26 | 3imtr4i 281 |
. . . . . 6
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → 𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑}) |
28 | | ne0i 3921 |
. . . . . 6
⊢ (𝑧 ∈ {𝑥 ∈ {𝑧} ∣ 𝜑} → {𝑥 ∈ {𝑧} ∣ 𝜑} ≠ ∅) |
29 | 27, 28 | syl 17 |
. . . . 5
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → {𝑥 ∈ {𝑧} ∣ 𝜑} ≠ ∅) |
30 | | rabn0 3958 |
. . . . 5
⊢ ({𝑥 ∈ {𝑧} ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ {𝑧}𝜑) |
31 | 29, 30 | sylib 208 |
. . . 4
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∃𝑥 ∈ {𝑧}𝜑) |
32 | | snex 4908 |
. . . . 5
⊢ {𝑧} ∈ V |
33 | | sseq1 3626 |
. . . . . 6
⊢ (𝑦 = {𝑧} → (𝑦 ⊆ 𝐴 ↔ {𝑧} ⊆ 𝐴)) |
34 | | rexeq 3139 |
. . . . . 6
⊢ (𝑦 = {𝑧} → (∃𝑥 ∈ 𝑦 𝜑 ↔ ∃𝑥 ∈ {𝑧}𝜑)) |
35 | 33, 34 | anbi12d 747 |
. . . . 5
⊢ (𝑦 = {𝑧} → ((𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑) ↔ ({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑))) |
36 | 32, 35 | spcev 3300 |
. . . 4
⊢ (({𝑧} ⊆ 𝐴 ∧ ∃𝑥 ∈ {𝑧}𝜑) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑)) |
37 | 11, 31, 36 | syl2anc 693 |
. . 3
⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑)) |
38 | 37 | exlimiv 1858 |
. 2
⊢
(∃𝑧 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑)) |
39 | 5, 38 | sylbi 207 |
1
⊢
(∃𝑥 ∈
𝐴 𝜑 → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑)) |