Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1533.1 |
⊢ ( 𝜃 → ∀ 𝑧 ∈ 𝐵 ¬ 𝑧 ∈ 𝐷 ) |
2 |
|
bnj1533.2 |
⊢ 𝐵 ⊆ 𝐴 |
3 |
|
bnj1533.3 |
⊢ 𝐷 = { 𝑧 ∈ 𝐴 ∣ 𝐶 ≠ 𝐸 } |
4 |
1
|
bnj1211 |
⊢ ( 𝜃 → ∀ 𝑧 ( 𝑧 ∈ 𝐵 → ¬ 𝑧 ∈ 𝐷 ) ) |
5 |
3
|
rabeq2i |
⊢ ( 𝑧 ∈ 𝐷 ↔ ( 𝑧 ∈ 𝐴 ∧ 𝐶 ≠ 𝐸 ) ) |
6 |
5
|
notbii |
⊢ ( ¬ 𝑧 ∈ 𝐷 ↔ ¬ ( 𝑧 ∈ 𝐴 ∧ 𝐶 ≠ 𝐸 ) ) |
7 |
|
imnan |
⊢ ( ( 𝑧 ∈ 𝐴 → ¬ 𝐶 ≠ 𝐸 ) ↔ ¬ ( 𝑧 ∈ 𝐴 ∧ 𝐶 ≠ 𝐸 ) ) |
8 |
|
nne |
⊢ ( ¬ 𝐶 ≠ 𝐸 ↔ 𝐶 = 𝐸 ) |
9 |
8
|
imbi2i |
⊢ ( ( 𝑧 ∈ 𝐴 → ¬ 𝐶 ≠ 𝐸 ) ↔ ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ) |
10 |
6 7 9
|
3bitr2i |
⊢ ( ¬ 𝑧 ∈ 𝐷 ↔ ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ) |
11 |
10
|
imbi2i |
⊢ ( ( 𝑧 ∈ 𝐵 → ¬ 𝑧 ∈ 𝐷 ) ↔ ( 𝑧 ∈ 𝐵 → ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ) ) |
12 |
2
|
sseli |
⊢ ( 𝑧 ∈ 𝐵 → 𝑧 ∈ 𝐴 ) |
13 |
12
|
imim1i |
⊢ ( ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) → ( 𝑧 ∈ 𝐵 → 𝐶 = 𝐸 ) ) |
14 |
|
ax-1 |
⊢ ( ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) → ( 𝑧 ∈ 𝐵 → ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ) ) |
15 |
14
|
anim1i |
⊢ ( ( ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑧 ∈ 𝐵 → ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ) ∧ 𝑧 ∈ 𝐵 ) ) |
16 |
|
simpr |
⊢ ( ( ( 𝑧 ∈ 𝐵 → ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ) ∧ 𝑧 ∈ 𝐵 ) → 𝑧 ∈ 𝐵 ) |
17 |
|
simpl |
⊢ ( ( ( 𝑧 ∈ 𝐵 → ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ) ∧ 𝑧 ∈ 𝐵 ) → ( 𝑧 ∈ 𝐵 → ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ) ) |
18 |
16 17
|
mpd |
⊢ ( ( ( 𝑧 ∈ 𝐵 → ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ) ∧ 𝑧 ∈ 𝐵 ) → ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ) |
19 |
18 16
|
jca |
⊢ ( ( ( 𝑧 ∈ 𝐵 → ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ) ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ∧ 𝑧 ∈ 𝐵 ) ) |
20 |
15 19
|
impbii |
⊢ ( ( ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ∧ 𝑧 ∈ 𝐵 ) ↔ ( ( 𝑧 ∈ 𝐵 → ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ) ∧ 𝑧 ∈ 𝐵 ) ) |
21 |
20
|
imbi1i |
⊢ ( ( ( ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ∧ 𝑧 ∈ 𝐵 ) → 𝐶 = 𝐸 ) ↔ ( ( ( 𝑧 ∈ 𝐵 → ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ) ∧ 𝑧 ∈ 𝐵 ) → 𝐶 = 𝐸 ) ) |
22 |
|
impexp |
⊢ ( ( ( ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ∧ 𝑧 ∈ 𝐵 ) → 𝐶 = 𝐸 ) ↔ ( ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) → ( 𝑧 ∈ 𝐵 → 𝐶 = 𝐸 ) ) ) |
23 |
|
impexp |
⊢ ( ( ( ( 𝑧 ∈ 𝐵 → ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ) ∧ 𝑧 ∈ 𝐵 ) → 𝐶 = 𝐸 ) ↔ ( ( 𝑧 ∈ 𝐵 → ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ) → ( 𝑧 ∈ 𝐵 → 𝐶 = 𝐸 ) ) ) |
24 |
21 22 23
|
3bitr3i |
⊢ ( ( ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) → ( 𝑧 ∈ 𝐵 → 𝐶 = 𝐸 ) ) ↔ ( ( 𝑧 ∈ 𝐵 → ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ) → ( 𝑧 ∈ 𝐵 → 𝐶 = 𝐸 ) ) ) |
25 |
13 24
|
mpbi |
⊢ ( ( 𝑧 ∈ 𝐵 → ( 𝑧 ∈ 𝐴 → 𝐶 = 𝐸 ) ) → ( 𝑧 ∈ 𝐵 → 𝐶 = 𝐸 ) ) |
26 |
11 25
|
sylbi |
⊢ ( ( 𝑧 ∈ 𝐵 → ¬ 𝑧 ∈ 𝐷 ) → ( 𝑧 ∈ 𝐵 → 𝐶 = 𝐸 ) ) |
27 |
4 26
|
sylg |
⊢ ( 𝜃 → ∀ 𝑧 ( 𝑧 ∈ 𝐵 → 𝐶 = 𝐸 ) ) |
28 |
27
|
bnj1142 |
⊢ ( 𝜃 → ∀ 𝑧 ∈ 𝐵 𝐶 = 𝐸 ) |