Step |
Hyp |
Ref |
Expression |
1 |
|
pm3.4 |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜑 → 𝜓 ) ) |
2 |
|
pm2.24 |
⊢ ( 𝜑 → ( ¬ 𝜑 → 𝜒 ) ) |
3 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ¬ 𝜑 → 𝜒 ) ) |
4 |
1 3
|
jca |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) ) |
5 |
|
pm2.21 |
⊢ ( ¬ 𝜑 → ( 𝜑 → 𝜓 ) ) |
6 |
5
|
adantr |
⊢ ( ( ¬ 𝜑 ∧ 𝜒 ) → ( 𝜑 → 𝜓 ) ) |
7 |
|
pm3.4 |
⊢ ( ( ¬ 𝜑 ∧ 𝜒 ) → ( ¬ 𝜑 → 𝜒 ) ) |
8 |
6 7
|
jca |
⊢ ( ( ¬ 𝜑 ∧ 𝜒 ) → ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) ) |
9 |
4 8
|
jaoi |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) ) |
10 |
|
pm2.27 |
⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) |
11 |
10
|
imdistani |
⊢ ( ( 𝜑 ∧ ( 𝜑 → 𝜓 ) ) → ( 𝜑 ∧ 𝜓 ) ) |
12 |
11
|
orcd |
⊢ ( ( 𝜑 ∧ ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜒 ) ) ) |
13 |
12
|
adantrr |
⊢ ( ( 𝜑 ∧ ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜒 ) ) ) |
14 |
|
pm2.27 |
⊢ ( ¬ 𝜑 → ( ( ¬ 𝜑 → 𝜒 ) → 𝜒 ) ) |
15 |
14
|
imdistani |
⊢ ( ( ¬ 𝜑 ∧ ( ¬ 𝜑 → 𝜒 ) ) → ( ¬ 𝜑 ∧ 𝜒 ) ) |
16 |
15
|
olcd |
⊢ ( ( ¬ 𝜑 ∧ ( ¬ 𝜑 → 𝜒 ) ) → ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜒 ) ) ) |
17 |
16
|
adantrl |
⊢ ( ( ¬ 𝜑 ∧ ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜒 ) ) ) |
18 |
13 17
|
pm2.61ian |
⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) → ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜒 ) ) ) |
19 |
9 18
|
impbii |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜒 ) ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) ) |