Step |
Hyp |
Ref |
Expression |
1 |
|
elpri |
⊢ ( 𝐴 ∈ { 𝐵 , 𝐶 } → ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) ) |
2 |
|
neeq1 |
⊢ ( 𝐵 = 𝐴 → ( 𝐵 ≠ 𝐶 ↔ 𝐴 ≠ 𝐶 ) ) |
3 |
2
|
eqcoms |
⊢ ( 𝐴 = 𝐵 → ( 𝐵 ≠ 𝐶 ↔ 𝐴 ≠ 𝐶 ) ) |
4 |
|
pm5.1 |
⊢ ( ( 𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶 ) → ( 𝐴 = 𝐵 ↔ 𝐴 ≠ 𝐶 ) ) |
5 |
4
|
ex |
⊢ ( 𝐴 = 𝐵 → ( 𝐴 ≠ 𝐶 → ( 𝐴 = 𝐵 ↔ 𝐴 ≠ 𝐶 ) ) ) |
6 |
3 5
|
sylbid |
⊢ ( 𝐴 = 𝐵 → ( 𝐵 ≠ 𝐶 → ( 𝐴 = 𝐵 ↔ 𝐴 ≠ 𝐶 ) ) ) |
7 |
|
neeq2 |
⊢ ( 𝐴 = 𝐶 → ( 𝐵 ≠ 𝐴 ↔ 𝐵 ≠ 𝐶 ) ) |
8 |
|
nesym |
⊢ ( 𝐵 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐵 ) |
9 |
|
pm5.1 |
⊢ ( ( 𝐴 = 𝐶 ∧ ¬ 𝐴 = 𝐵 ) → ( 𝐴 = 𝐶 ↔ ¬ 𝐴 = 𝐵 ) ) |
10 |
8 9
|
sylan2b |
⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 ≠ 𝐴 ) → ( 𝐴 = 𝐶 ↔ ¬ 𝐴 = 𝐵 ) ) |
11 |
10
|
necon2abid |
⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 ≠ 𝐴 ) → ( 𝐴 = 𝐵 ↔ 𝐴 ≠ 𝐶 ) ) |
12 |
11
|
ex |
⊢ ( 𝐴 = 𝐶 → ( 𝐵 ≠ 𝐴 → ( 𝐴 = 𝐵 ↔ 𝐴 ≠ 𝐶 ) ) ) |
13 |
7 12
|
sylbird |
⊢ ( 𝐴 = 𝐶 → ( 𝐵 ≠ 𝐶 → ( 𝐴 = 𝐵 ↔ 𝐴 ≠ 𝐶 ) ) ) |
14 |
6 13
|
jaoi |
⊢ ( ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) → ( 𝐵 ≠ 𝐶 → ( 𝐴 = 𝐵 ↔ 𝐴 ≠ 𝐶 ) ) ) |
15 |
1 14
|
syl |
⊢ ( 𝐴 ∈ { 𝐵 , 𝐶 } → ( 𝐵 ≠ 𝐶 → ( 𝐴 = 𝐵 ↔ 𝐴 ≠ 𝐶 ) ) ) |
16 |
15
|
imp |
⊢ ( ( 𝐴 ∈ { 𝐵 , 𝐶 } ∧ 𝐵 ≠ 𝐶 ) → ( 𝐴 = 𝐵 ↔ 𝐴 ≠ 𝐶 ) ) |