Step |
Hyp |
Ref |
Expression |
1 |
|
elpreq.1 |
⊢ ( 𝜑 → 𝑋 ∈ { 𝐴 , 𝐵 } ) |
2 |
|
elpreq.2 |
⊢ ( 𝜑 → 𝑌 ∈ { 𝐴 , 𝐵 } ) |
3 |
|
elpreq.3 |
⊢ ( 𝜑 → ( 𝑋 = 𝐴 ↔ 𝑌 = 𝐴 ) ) |
4 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝐴 ) → 𝑋 = 𝐴 ) |
5 |
3
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝐴 ) → 𝑌 = 𝐴 ) |
6 |
4 5
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝐴 ) → 𝑋 = 𝑌 ) |
7 |
|
elpri |
⊢ ( 𝑋 ∈ { 𝐴 , 𝐵 } → ( 𝑋 = 𝐴 ∨ 𝑋 = 𝐵 ) ) |
8 |
1 7
|
syl |
⊢ ( 𝜑 → ( 𝑋 = 𝐴 ∨ 𝑋 = 𝐵 ) ) |
9 |
8
|
orcanai |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 = 𝐴 ) → 𝑋 = 𝐵 ) |
10 |
|
simpl |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 = 𝐴 ) → 𝜑 ) |
11 |
3
|
notbid |
⊢ ( 𝜑 → ( ¬ 𝑋 = 𝐴 ↔ ¬ 𝑌 = 𝐴 ) ) |
12 |
11
|
biimpa |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 = 𝐴 ) → ¬ 𝑌 = 𝐴 ) |
13 |
|
elpri |
⊢ ( 𝑌 ∈ { 𝐴 , 𝐵 } → ( 𝑌 = 𝐴 ∨ 𝑌 = 𝐵 ) ) |
14 |
|
pm2.53 |
⊢ ( ( 𝑌 = 𝐴 ∨ 𝑌 = 𝐵 ) → ( ¬ 𝑌 = 𝐴 → 𝑌 = 𝐵 ) ) |
15 |
2 13 14
|
3syl |
⊢ ( 𝜑 → ( ¬ 𝑌 = 𝐴 → 𝑌 = 𝐵 ) ) |
16 |
10 12 15
|
sylc |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 = 𝐴 ) → 𝑌 = 𝐵 ) |
17 |
9 16
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ¬ 𝑋 = 𝐴 ) → 𝑋 = 𝑌 ) |
18 |
6 17
|
pm2.61dan |
⊢ ( 𝜑 → 𝑋 = 𝑌 ) |