Proof of Theorem elrab3t
Step | Hyp | Ref
| Expression |
1 | | simpr 108 |
. . 3
⊢
((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) |
2 | | nfa1 1474 |
. . . . 5
⊢
Ⅎ𝑥∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
3 | | nfv 1461 |
. . . . 5
⊢
Ⅎ𝑥 𝐴 ∈ 𝐵 |
4 | 2, 3 | nfan 1497 |
. . . 4
⊢
Ⅎ𝑥(∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) |
5 | | simpl 107 |
. . . . . 6
⊢
((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) |
6 | 5 | 19.21bi 1490 |
. . . . 5
⊢
((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) |
7 | | eleq1 2141 |
. . . . . . . . . 10
⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
8 | 7 | biimparc 293 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) → 𝑥 ∈ 𝐵) |
9 | 8 | biantrurd 299 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) → (𝜑 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))) |
10 | 9 | bibi1d 231 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝐵 ∧ 𝑥 = 𝐴) → ((𝜑 ↔ 𝜓) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓))) |
11 | 10 | pm5.74da 431 |
. . . . . 6
⊢ (𝐴 ∈ 𝐵 → ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ↔ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓)))) |
12 | 11 | adantl 271 |
. . . . 5
⊢
((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ↔ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓)))) |
13 | 6, 12 | mpbid 145 |
. . . 4
⊢
((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓))) |
14 | 4, 13 | alrimi 1455 |
. . 3
⊢
((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → ∀𝑥(𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓))) |
15 | | elabgt 2735 |
. . 3
⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ 𝜓)) |
16 | 1, 14, 15 | syl2anc 403 |
. 2
⊢
((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ 𝜓)) |
17 | | df-rab 2357 |
. . . 4
⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} |
18 | 17 | eleq2i 2145 |
. . 3
⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}) |
19 | 18 | bibi1i 226 |
. 2
⊢ ((𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓) ↔ (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ 𝜓)) |
20 | 16, 19 | sylibr 132 |
1
⊢
((∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝜓)) |