Step | Hyp | Ref
| Expression |
1 | | genp.1 |
. . . . . . . . . 10
⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)}) |
2 | | genp.2 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ Q ∧
𝑧 ∈ Q)
→ (𝑦𝐺𝑧) ∈ Q) |
3 | 1, 2 | genpelv 10766 |
. . . . . . . . 9
⊢ ((𝐵 ∈ P ∧
𝐶 ∈ P)
→ (𝑡 ∈ (𝐵𝐹𝐶) ↔ ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ))) |
4 | 3 | 3adant1 1129 |
. . . . . . . 8
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P
∧ 𝐶 ∈
P) → (𝑡
∈ (𝐵𝐹𝐶) ↔ ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ))) |
5 | 4 | anbi1d 630 |
. . . . . . 7
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P
∧ 𝐶 ∈
P) → ((𝑡
∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))) |
6 | 5 | exbidv 1924 |
. . . . . 6
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P
∧ 𝐶 ∈
P) → (∃𝑡(𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃𝑡(∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)))) |
7 | | df-rex 3070 |
. . . . . 6
⊢
(∃𝑡 ∈
(𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑡(𝑡 ∈ (𝐵𝐹𝐶) ∧ 𝑥 = (𝑓𝐺𝑡))) |
8 | | ovex 7300 |
. . . . . . . . . . . . 13
⊢ (𝑔𝐺ℎ) ∈ V |
9 | 8 | isseti 3444 |
. . . . . . . . . . . 12
⊢
∃𝑡 𝑡 = (𝑔𝐺ℎ) |
10 | 9 | biantrur 531 |
. . . . . . . . . . 11
⊢ (𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ (∃𝑡 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
11 | | 19.41v 1953 |
. . . . . . . . . . 11
⊢
(∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑡 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
12 | 10, 11 | bitr4i 277 |
. . . . . . . . . 10
⊢ (𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
13 | 12 | rexbii 3179 |
. . . . . . . . 9
⊢
(∃ℎ ∈
𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃ℎ ∈ 𝐶 ∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
14 | | rexcom4 3231 |
. . . . . . . . 9
⊢
(∃ℎ ∈
𝐶 ∃𝑡(𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑡∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
15 | 13, 14 | bitri 274 |
. . . . . . . 8
⊢
(∃ℎ ∈
𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
16 | 15 | rexbii 3179 |
. . . . . . 7
⊢
(∃𝑔 ∈
𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑔 ∈ 𝐵 ∃𝑡∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
17 | | rexcom4 3231 |
. . . . . . 7
⊢
(∃𝑔 ∈
𝐵 ∃𝑡∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑡∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
18 | | oveq2 7275 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = (𝑔𝐺ℎ) → (𝑓𝐺𝑡) = (𝑓𝐺(𝑔𝐺ℎ))) |
19 | | genpass.6 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ)) |
20 | 18, 19 | eqtr4di 2796 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = (𝑔𝐺ℎ) → (𝑓𝐺𝑡) = ((𝑓𝐺𝑔)𝐺ℎ)) |
21 | 20 | eqeq2d 2749 |
. . . . . . . . . . . . 13
⊢ (𝑡 = (𝑔𝐺ℎ) → (𝑥 = (𝑓𝐺𝑡) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
22 | 21 | pm5.32i 575 |
. . . . . . . . . . . 12
⊢ ((𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
23 | 22 | rexbii 3179 |
. . . . . . . . . . 11
⊢
(∃ℎ ∈
𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ ∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
24 | | r19.41v 3274 |
. . . . . . . . . . 11
⊢
(∃ℎ ∈
𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))) |
25 | 23, 24 | bitr3i 276 |
. . . . . . . . . 10
⊢
(∃ℎ ∈
𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))) |
26 | 25 | rexbii 3179 |
. . . . . . . . 9
⊢
(∃𝑔 ∈
𝐵 ∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑔 ∈ 𝐵 (∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))) |
27 | | r19.41v 3274 |
. . . . . . . . 9
⊢
(∃𝑔 ∈
𝐵 (∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡)) ↔ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))) |
28 | 26, 27 | bitri 274 |
. . . . . . . 8
⊢
(∃𝑔 ∈
𝐵 ∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))) |
29 | 28 | exbii 1850 |
. . . . . . 7
⊢
(∃𝑡∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 (𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ ∃𝑡(∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))) |
30 | 16, 17, 29 | 3bitri 297 |
. . . . . 6
⊢
(∃𝑔 ∈
𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑡 = (𝑔𝐺ℎ) ∧ 𝑥 = (𝑓𝐺𝑡))) |
31 | 6, 7, 30 | 3bitr4g 314 |
. . . . 5
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P
∧ 𝐶 ∈
P) → (∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
32 | 31 | rexbidv 3224 |
. . . 4
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P
∧ 𝐶 ∈
P) → (∃𝑓 ∈ 𝐴 ∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡) ↔ ∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
33 | | genpass.5 |
. . . . . . 7
⊢ ((𝑓 ∈ P ∧
𝑔 ∈ P)
→ (𝑓𝐹𝑔) ∈ P) |
34 | 33 | caovcl 7456 |
. . . . . 6
⊢ ((𝐵 ∈ P ∧
𝐶 ∈ P)
→ (𝐵𝐹𝐶) ∈ P) |
35 | 1, 2 | genpelv 10766 |
. . . . . 6
⊢ ((𝐴 ∈ P ∧
(𝐵𝐹𝐶) ∈ P) → (𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓 ∈ 𝐴 ∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡))) |
36 | 34, 35 | sylan2 593 |
. . . . 5
⊢ ((𝐴 ∈ P ∧
(𝐵 ∈ P
∧ 𝐶 ∈
P)) → (𝑥
∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓 ∈ 𝐴 ∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡))) |
37 | 36 | 3impb 1114 |
. . . 4
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P
∧ 𝐶 ∈
P) → (𝑥
∈ (𝐴𝐹(𝐵𝐹𝐶)) ↔ ∃𝑓 ∈ 𝐴 ∃𝑡 ∈ (𝐵𝐹𝐶)𝑥 = (𝑓𝐺𝑡))) |
38 | 33 | caovcl 7456 |
. . . . . 6
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝐴𝐹𝐵) ∈ P) |
39 | 1, 2 | genpelv 10766 |
. . . . . 6
⊢ (((𝐴𝐹𝐵) ∈ P ∧ 𝐶 ∈ P) →
(𝑥 ∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑡 ∈ (𝐴𝐹𝐵)∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ))) |
40 | 38, 39 | stoic3 1779 |
. . . . 5
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P
∧ 𝐶 ∈
P) → (𝑥
∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑡 ∈ (𝐴𝐹𝐵)∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ))) |
41 | 1, 2 | genpelv 10766 |
. . . . . . . . 9
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P)
→ (𝑡 ∈ (𝐴𝐹𝐵) ↔ ∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔))) |
42 | 41 | 3adant3 1131 |
. . . . . . . 8
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P
∧ 𝐶 ∈
P) → (𝑡
∈ (𝐴𝐹𝐵) ↔ ∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔))) |
43 | 42 | anbi1d 630 |
. . . . . . 7
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P
∧ 𝐶 ∈
P) → ((𝑡
∈ (𝐴𝐹𝐵) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ (∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))) |
44 | 43 | exbidv 1924 |
. . . . . 6
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P
∧ 𝐶 ∈
P) → (∃𝑡(𝑡 ∈ (𝐴𝐹𝐵) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡(∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)))) |
45 | | df-rex 3070 |
. . . . . 6
⊢
(∃𝑡 ∈
(𝐴𝐹𝐵)∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ) ↔ ∃𝑡(𝑡 ∈ (𝐴𝐹𝐵) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ))) |
46 | | 19.41v 1953 |
. . . . . . . . . . 11
⊢
(∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ)) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
47 | | oveq1 7274 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = (𝑓𝐺𝑔) → (𝑡𝐺ℎ) = ((𝑓𝐺𝑔)𝐺ℎ)) |
48 | 47 | eqeq2d 2749 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = (𝑓𝐺𝑔) → (𝑥 = (𝑡𝐺ℎ) ↔ 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
49 | 48 | rexbidv 3224 |
. . . . . . . . . . . . 13
⊢ (𝑡 = (𝑓𝐺𝑔) → (∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ) ↔ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
50 | 49 | pm5.32i 575 |
. . . . . . . . . . . 12
⊢ ((𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
51 | 50 | exbii 1850 |
. . . . . . . . . . 11
⊢
(∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
52 | | ovex 7300 |
. . . . . . . . . . . . 13
⊢ (𝑓𝐺𝑔) ∈ V |
53 | 52 | isseti 3444 |
. . . . . . . . . . . 12
⊢
∃𝑡 𝑡 = (𝑓𝐺𝑔) |
54 | 53 | biantrur 531 |
. . . . . . . . . . 11
⊢
(∃ℎ ∈
𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ (∃𝑡 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
55 | 46, 51, 54 | 3bitr4ri 304 |
. . . . . . . . . 10
⊢
(∃ℎ ∈
𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ))) |
56 | 55 | rexbii 3179 |
. . . . . . . . 9
⊢
(∃𝑔 ∈
𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑔 ∈ 𝐵 ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ))) |
57 | | rexcom4 3231 |
. . . . . . . . 9
⊢
(∃𝑔 ∈
𝐵 ∃𝑡(𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ))) |
58 | 56, 57 | bitri 274 |
. . . . . . . 8
⊢
(∃𝑔 ∈
𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ))) |
59 | 58 | rexbii 3179 |
. . . . . . 7
⊢
(∃𝑓 ∈
𝐴 ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑓 ∈ 𝐴 ∃𝑡∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ))) |
60 | | rexcom4 3231 |
. . . . . . 7
⊢
(∃𝑓 ∈
𝐴 ∃𝑡∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ))) |
61 | | r19.41vv 3276 |
. . . . . . . 8
⊢
(∃𝑓 ∈
𝐴 ∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ (∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ))) |
62 | 61 | exbii 1850 |
. . . . . . 7
⊢
(∃𝑡∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 (𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ)) ↔ ∃𝑡(∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ))) |
63 | 59, 60, 62 | 3bitri 297 |
. . . . . 6
⊢
(∃𝑓 ∈
𝐴 ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ) ↔ ∃𝑡(∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 𝑡 = (𝑓𝐺𝑔) ∧ ∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ))) |
64 | 44, 45, 63 | 3bitr4g 314 |
. . . . 5
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P
∧ 𝐶 ∈
P) → (∃𝑡 ∈ (𝐴𝐹𝐵)∃ℎ ∈ 𝐶 𝑥 = (𝑡𝐺ℎ) ↔ ∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
65 | 40, 64 | bitrd 278 |
. . . 4
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P
∧ 𝐶 ∈
P) → (𝑥
∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ ∃𝑓 ∈ 𝐴 ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐶 𝑥 = ((𝑓𝐺𝑔)𝐺ℎ))) |
66 | 32, 37, 65 | 3bitr4rd 312 |
. . 3
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P
∧ 𝐶 ∈
P) → (𝑥
∈ ((𝐴𝐹𝐵)𝐹𝐶) ↔ 𝑥 ∈ (𝐴𝐹(𝐵𝐹𝐶)))) |
67 | 66 | eqrdv 2736 |
. 2
⊢ ((𝐴 ∈ P ∧
𝐵 ∈ P
∧ 𝐶 ∈
P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
68 | | genpass.4 |
. . 3
⊢ dom 𝐹 = (P ×
P) |
69 | | 0npr 10758 |
. . 3
⊢ ¬
∅ ∈ P |
70 | 68, 69 | ndmovass 7450 |
. 2
⊢ (¬
(𝐴 ∈ P
∧ 𝐵 ∈
P ∧ 𝐶
∈ P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
71 | 67, 70 | pm2.61i 182 |
1
⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)) |