Step |
Hyp |
Ref |
Expression |
1 |
|
vtocl3gaf.a |
⊢ Ⅎ 𝑥 𝐴 |
2 |
|
vtocl3gaf.b |
⊢ Ⅎ 𝑦 𝐴 |
3 |
|
vtocl3gaf.c |
⊢ Ⅎ 𝑧 𝐴 |
4 |
|
vtocl3gaf.d |
⊢ Ⅎ 𝑦 𝐵 |
5 |
|
vtocl3gaf.e |
⊢ Ⅎ 𝑧 𝐵 |
6 |
|
vtocl3gaf.f |
⊢ Ⅎ 𝑧 𝐶 |
7 |
|
vtocl3gaf.1 |
⊢ Ⅎ 𝑥 𝜓 |
8 |
|
vtocl3gaf.2 |
⊢ Ⅎ 𝑦 𝜒 |
9 |
|
vtocl3gaf.3 |
⊢ Ⅎ 𝑧 𝜃 |
10 |
|
vtocl3gaf.4 |
⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) |
11 |
|
vtocl3gaf.5 |
⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) ) |
12 |
|
vtocl3gaf.6 |
⊢ ( 𝑧 = 𝐶 → ( 𝜒 ↔ 𝜃 ) ) |
13 |
|
vtocl3gaf.7 |
⊢ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) → 𝜑 ) |
14 |
3
|
nfel1 |
⊢ Ⅎ 𝑧 𝐴 ∈ 𝑅 |
15 |
5
|
nfel1 |
⊢ Ⅎ 𝑧 𝐵 ∈ 𝑆 |
16 |
14 15
|
nfan |
⊢ Ⅎ 𝑧 ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) |
17 |
16 9
|
nfim |
⊢ Ⅎ 𝑧 ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → 𝜃 ) |
18 |
12
|
imbi2d |
⊢ ( 𝑧 = 𝐶 → ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → 𝜒 ) ↔ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → 𝜃 ) ) ) |
19 |
|
nfv |
⊢ Ⅎ 𝑥 𝑧 ∈ 𝑇 |
20 |
19 7
|
nfim |
⊢ Ⅎ 𝑥 ( 𝑧 ∈ 𝑇 → 𝜓 ) |
21 |
|
nfv |
⊢ Ⅎ 𝑦 𝑧 ∈ 𝑇 |
22 |
21 8
|
nfim |
⊢ Ⅎ 𝑦 ( 𝑧 ∈ 𝑇 → 𝜒 ) |
23 |
10
|
imbi2d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑧 ∈ 𝑇 → 𝜑 ) ↔ ( 𝑧 ∈ 𝑇 → 𝜓 ) ) ) |
24 |
11
|
imbi2d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝑧 ∈ 𝑇 → 𝜓 ) ↔ ( 𝑧 ∈ 𝑇 → 𝜒 ) ) ) |
25 |
13
|
3expia |
⊢ ( ( 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ) → ( 𝑧 ∈ 𝑇 → 𝜑 ) ) |
26 |
1 2 4 20 22 23 24 25
|
vtocl2gaf |
⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( 𝑧 ∈ 𝑇 → 𝜒 ) ) |
27 |
26
|
com12 |
⊢ ( 𝑧 ∈ 𝑇 → ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → 𝜒 ) ) |
28 |
6 17 18 27
|
vtoclgaf |
⊢ ( 𝐶 ∈ 𝑇 → ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → 𝜃 ) ) |
29 |
28
|
impcom |
⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ 𝐶 ∈ 𝑇 ) → 𝜃 ) |
30 |
29
|
3impa |
⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇 ) → 𝜃 ) |