Step | Hyp | Ref
| Expression |
1 | | ssid 3942 |
. 2
⊢ 𝑇 ⊆ 𝑇 |
2 | | eqid 2738 |
. . . . 5
⊢ 𝑇 = 𝑇 |
3 | | tfisi.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
4 | | tfisi.b |
. . . . . . 7
⊢ (𝜑 → 𝑇 ∈ On) |
5 | | eqeq2 2750 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑤 → (𝑅 = 𝑧 ↔ 𝑅 = 𝑤)) |
6 | | sseq1 3945 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑤 → (𝑧 ⊆ 𝑇 ↔ 𝑤 ⊆ 𝑇)) |
7 | 6 | anbi2d 629 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑤 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) ↔ (𝜑 ∧ 𝑤 ⊆ 𝑇))) |
8 | 7 | imbi1d 342 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑤 → (((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓) ↔ ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜓))) |
9 | 5, 8 | imbi12d 345 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑤 → ((𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)) ↔ (𝑅 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜓)))) |
10 | 9 | albidv 1923 |
. . . . . . . . 9
⊢ (𝑧 = 𝑤 → (∀𝑥(𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)) ↔ ∀𝑥(𝑅 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜓)))) |
11 | | tfisi.f |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → 𝑅 = 𝑆) |
12 | 11 | eqeq1d 2740 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑅 = 𝑤 ↔ 𝑆 = 𝑤)) |
13 | | tfisi.d |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) |
14 | 13 | imbi2d 341 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜓) ↔ ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) |
15 | 12, 14 | imbi12d 345 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → ((𝑅 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜓)) ↔ (𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒)))) |
16 | 15 | cbvalvw 2039 |
. . . . . . . . 9
⊢
(∀𝑥(𝑅 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜓)) ↔ ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) |
17 | 10, 16 | bitrdi 287 |
. . . . . . . 8
⊢ (𝑧 = 𝑤 → (∀𝑥(𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)) ↔ ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒)))) |
18 | | eqeq2 2750 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑇 → (𝑅 = 𝑧 ↔ 𝑅 = 𝑇)) |
19 | | sseq1 3945 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑇 → (𝑧 ⊆ 𝑇 ↔ 𝑇 ⊆ 𝑇)) |
20 | 19 | anbi2d 629 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑇 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) ↔ (𝜑 ∧ 𝑇 ⊆ 𝑇))) |
21 | 20 | imbi1d 342 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑇 → (((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓) ↔ ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓))) |
22 | 18, 21 | imbi12d 345 |
. . . . . . . . 9
⊢ (𝑧 = 𝑇 → ((𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)) ↔ (𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓)))) |
23 | 22 | albidv 1923 |
. . . . . . . 8
⊢ (𝑧 = 𝑇 → (∀𝑥(𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)) ↔ ∀𝑥(𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓)))) |
24 | | simp3l 1200 |
. . . . . . . . . . . 12
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝜑) |
25 | | simp2 1136 |
. . . . . . . . . . . . 13
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝑅 = 𝑧) |
26 | | simp1l 1196 |
. . . . . . . . . . . . 13
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝑧 ∈ On) |
27 | 25, 26 | eqeltrd 2839 |
. . . . . . . . . . . 12
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝑅 ∈ On) |
28 | | simp3r 1201 |
. . . . . . . . . . . . 13
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝑧 ⊆ 𝑇) |
29 | 25, 28 | eqsstrd 3958 |
. . . . . . . . . . . 12
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝑅 ⊆ 𝑇) |
30 | | simpl3l 1227 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → 𝜑) |
31 | | simpl1l 1223 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → 𝑧 ∈ On) |
32 | | simpr 485 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) |
33 | | simpl2 1191 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → 𝑅 = 𝑧) |
34 | 32, 33 | eleqtrd 2841 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑧) |
35 | | onelss 6301 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ On →
(⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑧 → ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑧)) |
36 | 31, 34, 35 | sylc 65 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑧) |
37 | | simpl3r 1228 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → 𝑧 ⊆ 𝑇) |
38 | 36, 37 | sstrd 3930 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) |
39 | | eqeq2 2750 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → (𝑆 = 𝑤 ↔ 𝑆 = ⦋𝑣 / 𝑥⦌𝑅)) |
40 | | sseq1 3945 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → (𝑤 ⊆ 𝑇 ↔ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇)) |
41 | 40 | anbi2d 629 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) ↔ (𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇))) |
42 | 41 | imbi1d 342 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → (((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒) ↔ ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒))) |
43 | 39, 42 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒)) ↔ (𝑆 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒)))) |
44 | 43 | albidv 1923 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → (∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒)) ↔ ∀𝑦(𝑆 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒)))) |
45 | | simpl1r 1224 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) |
46 | 44, 45, 34 | rspcdva 3561 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → ∀𝑦(𝑆 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒))) |
47 | | eqidd 2739 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → ⦋𝑣 / 𝑥⦌𝑅 = ⦋𝑣 / 𝑥⦌𝑅) |
48 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑥𝑦 |
49 | | nfcv 2907 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑥𝑆 |
50 | 48, 49, 11 | csbhypf 3860 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 = 𝑦 → ⦋𝑣 / 𝑥⦌𝑅 = 𝑆) |
51 | 50 | eqcomd 2744 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 = 𝑦 → 𝑆 = ⦋𝑣 / 𝑥⦌𝑅) |
52 | 51 | equcoms 2023 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑣 → 𝑆 = ⦋𝑣 / 𝑥⦌𝑅) |
53 | 52 | eqeq1d 2740 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 𝑣 → (𝑆 = ⦋𝑣 / 𝑥⦌𝑅 ↔ ⦋𝑣 / 𝑥⦌𝑅 = ⦋𝑣 / 𝑥⦌𝑅)) |
54 | | nfv 1917 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑥𝜒 |
55 | 54, 13 | sbhypf 3488 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 = 𝑦 → ([𝑣 / 𝑥]𝜓 ↔ 𝜒)) |
56 | 55 | bicomd 222 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 = 𝑦 → (𝜒 ↔ [𝑣 / 𝑥]𝜓)) |
57 | 56 | equcoms 2023 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑣 → (𝜒 ↔ [𝑣 / 𝑥]𝜓)) |
58 | 57 | imbi2d 341 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 𝑣 → (((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒) ↔ ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → [𝑣 / 𝑥]𝜓))) |
59 | 53, 58 | imbi12d 345 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 = 𝑣 → ((𝑆 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒)) ↔ (⦋𝑣 / 𝑥⦌𝑅 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → [𝑣 / 𝑥]𝜓)))) |
60 | 59 | spvv 2000 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑦(𝑆 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒)) → (⦋𝑣 / 𝑥⦌𝑅 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → [𝑣 / 𝑥]𝜓))) |
61 | 46, 47, 60 | sylc 65 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → [𝑣 / 𝑥]𝜓)) |
62 | 30, 38, 61 | mp2and 696 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → [𝑣 / 𝑥]𝜓) |
63 | 62 | ex 413 |
. . . . . . . . . . . . . 14
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → (⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅 → [𝑣 / 𝑥]𝜓)) |
64 | 63 | alrimiv 1930 |
. . . . . . . . . . . . 13
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → ∀𝑣(⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅 → [𝑣 / 𝑥]𝜓)) |
65 | 50 | eleq1d 2823 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 = 𝑦 → (⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅 ↔ 𝑆 ∈ 𝑅)) |
66 | 65, 55 | imbi12d 345 |
. . . . . . . . . . . . . 14
⊢ (𝑣 = 𝑦 → ((⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅 → [𝑣 / 𝑥]𝜓) ↔ (𝑆 ∈ 𝑅 → 𝜒))) |
67 | 66 | cbvalvw 2039 |
. . . . . . . . . . . . 13
⊢
(∀𝑣(⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅 → [𝑣 / 𝑥]𝜓) ↔ ∀𝑦(𝑆 ∈ 𝑅 → 𝜒)) |
68 | 64, 67 | sylib 217 |
. . . . . . . . . . . 12
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → ∀𝑦(𝑆 ∈ 𝑅 → 𝜒)) |
69 | | tfisi.c |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑅 ∈ On ∧ 𝑅 ⊆ 𝑇) ∧ ∀𝑦(𝑆 ∈ 𝑅 → 𝜒)) → 𝜓) |
70 | 24, 27, 29, 68, 69 | syl121anc 1374 |
. . . . . . . . . . 11
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝜓) |
71 | 70 | 3exp 1118 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) → (𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓))) |
72 | 71 | alrimiv 1930 |
. . . . . . . . 9
⊢ ((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) → ∀𝑥(𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓))) |
73 | 72 | ex 413 |
. . . . . . . 8
⊢ (𝑧 ∈ On → (∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒)) → ∀𝑥(𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)))) |
74 | 17, 23, 73 | tfis3 7694 |
. . . . . . 7
⊢ (𝑇 ∈ On → ∀𝑥(𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓))) |
75 | 4, 74 | syl 17 |
. . . . . 6
⊢ (𝜑 → ∀𝑥(𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓))) |
76 | | tfisi.g |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → 𝑅 = 𝑇) |
77 | 76 | eqeq1d 2740 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → (𝑅 = 𝑇 ↔ 𝑇 = 𝑇)) |
78 | | tfisi.e |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) |
79 | 78 | imbi2d 341 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → (((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓) ↔ ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜃))) |
80 | 77, 79 | imbi12d 345 |
. . . . . . 7
⊢ (𝑥 = 𝐴 → ((𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓)) ↔ (𝑇 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜃)))) |
81 | 80 | spcgv 3532 |
. . . . . 6
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓)) → (𝑇 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜃)))) |
82 | 3, 75, 81 | sylc 65 |
. . . . 5
⊢ (𝜑 → (𝑇 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜃))) |
83 | 2, 82 | mpi 20 |
. . . 4
⊢ (𝜑 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜃)) |
84 | 83 | expd 416 |
. . 3
⊢ (𝜑 → (𝜑 → (𝑇 ⊆ 𝑇 → 𝜃))) |
85 | 84 | pm2.43i 52 |
. 2
⊢ (𝜑 → (𝑇 ⊆ 𝑇 → 𝜃)) |
86 | 1, 85 | mpi 20 |
1
⊢ (𝜑 → 𝜃) |