Step | Hyp | Ref
| Expression |
1 | | ssid 2964 |
. 2
⊢ 𝑇 ⊆ 𝑇 |
2 | | eqid 2040 |
. . . . 5
⊢ 𝑇 = 𝑇 |
3 | | tfisi.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
4 | | tfisi.b |
. . . . . . 7
⊢ (𝜑 → 𝑇 ∈ On) |
5 | | eqeq2 2049 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑤 → (𝑅 = 𝑧 ↔ 𝑅 = 𝑤)) |
6 | | sseq1 2966 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑤 → (𝑧 ⊆ 𝑇 ↔ 𝑤 ⊆ 𝑇)) |
7 | 6 | anbi2d 437 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑤 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) ↔ (𝜑 ∧ 𝑤 ⊆ 𝑇))) |
8 | 7 | imbi1d 220 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑤 → (((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓) ↔ ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜓))) |
9 | 5, 8 | imbi12d 223 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑤 → ((𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)) ↔ (𝑅 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜓)))) |
10 | 9 | albidv 1705 |
. . . . . . . . 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 2048 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑅 = 𝑤 ↔ 𝑆 = 𝑤)) |
13 | | tfisi.d |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) |
14 | 13 | imbi2d 219 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜓) ↔ ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) |
15 | 12, 14 | imbi12d 223 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → ((𝑅 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜓)) ↔ (𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒)))) |
16 | 15 | cbvalv 1794 |
. . . . . . . . 9
⊢
(∀𝑥(𝑅 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜓)) ↔ ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) |
17 | 10, 16 | syl6bb 185 |
. . . . . . . 8
⊢ (𝑧 = 𝑤 → (∀𝑥(𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)) ↔ ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒)))) |
18 | | eqeq2 2049 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑇 → (𝑅 = 𝑧 ↔ 𝑅 = 𝑇)) |
19 | | sseq1 2966 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑇 → (𝑧 ⊆ 𝑇 ↔ 𝑇 ⊆ 𝑇)) |
20 | 19 | anbi2d 437 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑇 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) ↔ (𝜑 ∧ 𝑇 ⊆ 𝑇))) |
21 | 20 | imbi1d 220 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑇 → (((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓) ↔ ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓))) |
22 | 18, 21 | imbi12d 223 |
. . . . . . . . 9
⊢ (𝑧 = 𝑇 → ((𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)) ↔ (𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓)))) |
23 | 22 | albidv 1705 |
. . . . . . . 8
⊢ (𝑧 = 𝑇 → (∀𝑥(𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)) ↔ ∀𝑥(𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓)))) |
24 | | simp3l 932 |
. . . . . . . . . . . 12
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝜑) |
25 | | simp2 905 |
. . . . . . . . . . . . 13
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝑅 = 𝑧) |
26 | | simp1l 928 |
. . . . . . . . . . . . 13
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝑧 ∈ On) |
27 | 25, 26 | eqeltrd 2114 |
. . . . . . . . . . . 12
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝑅 ∈ On) |
28 | | simp3r 933 |
. . . . . . . . . . . . 13
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝑧 ⊆ 𝑇) |
29 | 25, 28 | eqsstrd 2979 |
. . . . . . . . . . . 12
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝑅 ⊆ 𝑇) |
30 | | simpl3l 959 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → 𝜑) |
31 | | simpl1l 955 |
. . . . . . . . . . . . . . . . . 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 103 |
. . . . . . . . . . . . . . . . . . 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 908 |
. . . . . . . . . . . . . . . . . . 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 2116 |
. . . . . . . . . . . . . . . . . 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 4124 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ On →
(⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑧 → ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑧)) |
36 | 31, 34, 35 | sylc 56 |
. . . . . . . . . . . . . . . . 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 960 |
. . . . . . . . . . . . . . . . 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 2955 |
. . . . . . . . . . . . . . . 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 | | simpl1r 956 |
. . . . . . . . . . . . . . . . . 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))) |
40 | | eqeq2 2049 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → (𝑆 = 𝑤 ↔ 𝑆 = ⦋𝑣 / 𝑥⦌𝑅)) |
41 | | sseq1 2966 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → (𝑤 ⊆ 𝑇 ↔ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇)) |
42 | 41 | anbi2d 437 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) ↔ (𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇))) |
43 | 42 | imbi1d 220 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → (((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒) ↔ ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒))) |
44 | 40, 43 | imbi12d 223 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒)) ↔ (𝑆 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒)))) |
45 | 44 | albidv 1705 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → (∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒)) ↔ ∀𝑦(𝑆 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒)))) |
46 | 45 | rspcva 2654 |
. . . . . . . . . . . . . . . . . 18
⊢
((⦋𝑣 /
𝑥⦌𝑅 ∈ 𝑧 ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) → ∀𝑦(𝑆 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒))) |
47 | 34, 39, 46 | syl2anc 391 |
. . . . . . . . . . . . . . . . 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))) |
48 | | eqidd 2041 |
. . . . . . . . . . . . . . . . 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) |
49 | | nfcv 2178 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑥𝑦 |
50 | | nfcv 2178 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑥𝑆 |
51 | 49, 50, 11 | csbhypf 2885 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 = 𝑦 → ⦋𝑣 / 𝑥⦌𝑅 = 𝑆) |
52 | 51 | eqcomd 2045 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 = 𝑦 → 𝑆 = ⦋𝑣 / 𝑥⦌𝑅) |
53 | 52 | equcoms 1594 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑣 → 𝑆 = ⦋𝑣 / 𝑥⦌𝑅) |
54 | 53 | eqeq1d 2048 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 𝑣 → (𝑆 = ⦋𝑣 / 𝑥⦌𝑅 ↔ ⦋𝑣 / 𝑥⦌𝑅 = ⦋𝑣 / 𝑥⦌𝑅)) |
55 | | nfv 1421 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑥𝜒 |
56 | 55, 13 | sbhypf 2603 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 = 𝑦 → ([𝑣 / 𝑥]𝜓 ↔ 𝜒)) |
57 | 56 | bicomd 129 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 = 𝑦 → (𝜒 ↔ [𝑣 / 𝑥]𝜓)) |
58 | 57 | equcoms 1594 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑣 → (𝜒 ↔ [𝑣 / 𝑥]𝜓)) |
59 | 58 | imbi2d 219 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 𝑣 → (((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒) ↔ ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → [𝑣 / 𝑥]𝜓))) |
60 | 54, 59 | imbi12d 223 |
. . . . . . . . . . . . . . . . . 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)))) |
61 | 60 | spv 1740 |
. . . . . . . . . . . . . . . . 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))) |
62 | 47, 48, 61 | sylc 56 |
. . . . . . . . . . . . . . . 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)) |
63 | 30, 38, 62 | mp2and 409 |
. . . . . . . . . . . . . . 15
⊢ ((((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 | ex 108 |
. . . . . . . . . . . . . 14
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → (⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅 → [𝑣 / 𝑥]𝜓)) |
65 | 64 | alrimiv 1754 |
. . . . . . . . . . . . 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)) |
66 | 51 | eleq1d 2106 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 = 𝑦 → (⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅 ↔ 𝑆 ∈ 𝑅)) |
67 | 66, 56 | imbi12d 223 |
. . . . . . . . . . . . . 14
⊢ (𝑣 = 𝑦 → ((⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅 → [𝑣 / 𝑥]𝜓) ↔ (𝑆 ∈ 𝑅 → 𝜒))) |
68 | 67 | cbvalv 1794 |
. . . . . . . . . . . . 13
⊢
(∀𝑣(⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅 → [𝑣 / 𝑥]𝜓) ↔ ∀𝑦(𝑆 ∈ 𝑅 → 𝜒)) |
69 | 65, 68 | sylib 127 |
. . . . . . . . . . . 12
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → ∀𝑦(𝑆 ∈ 𝑅 → 𝜒)) |
70 | | tfisi.c |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑅 ∈ On ∧ 𝑅 ⊆ 𝑇) ∧ ∀𝑦(𝑆 ∈ 𝑅 → 𝜒)) → 𝜓) |
71 | 24, 27, 29, 69, 70 | syl121anc 1140 |
. . . . . . . . . . 11
⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝜓) |
72 | 71 | 3exp 1103 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) → (𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓))) |
73 | 72 | alrimiv 1754 |
. . . . . . . . 9
⊢ ((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) → ∀𝑥(𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓))) |
74 | 73 | ex 108 |
. . . . . . . 8
⊢ (𝑧 ∈ On → (∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒)) → ∀𝑥(𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)))) |
75 | 17, 23, 74 | tfis3 4309 |
. . . . . . 7
⊢ (𝑇 ∈ On → ∀𝑥(𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓))) |
76 | 4, 75 | syl 14 |
. . . . . 6
⊢ (𝜑 → ∀𝑥(𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓))) |
77 | | tfisi.g |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → 𝑅 = 𝑇) |
78 | 77 | eqeq1d 2048 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → (𝑅 = 𝑇 ↔ 𝑇 = 𝑇)) |
79 | | tfisi.e |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) |
80 | 79 | imbi2d 219 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → (((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓) ↔ ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜃))) |
81 | 78, 80 | imbi12d 223 |
. . . . . . 7
⊢ (𝑥 = 𝐴 → ((𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓)) ↔ (𝑇 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜃)))) |
82 | 81 | spcgv 2640 |
. . . . . 6
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓)) → (𝑇 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜃)))) |
83 | 3, 76, 82 | sylc 56 |
. . . . 5
⊢ (𝜑 → (𝑇 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜃))) |
84 | 2, 83 | mpi 15 |
. . . 4
⊢ (𝜑 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜃)) |
85 | 84 | expd 245 |
. . 3
⊢ (𝜑 → (𝜑 → (𝑇 ⊆ 𝑇 → 𝜃))) |
86 | 85 | pm2.43i 43 |
. 2
⊢ (𝜑 → (𝑇 ⊆ 𝑇 → 𝜃)) |
87 | 1, 86 | mpi 15 |
1
⊢ (𝜑 → 𝜃) |