Step |
Hyp |
Ref |
Expression |
1 |
|
ianor |
⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) ↔ ( ¬ 𝜑 ∨ ¬ 𝜓 ) ) |
2 |
|
ianor |
⊢ ( ¬ ( 𝜑 ∧ 𝜒 ) ↔ ( ¬ 𝜑 ∨ ¬ 𝜒 ) ) |
3 |
|
ianor |
⊢ ( ¬ ( 𝜓 ∧ 𝜒 ) ↔ ( ¬ 𝜓 ∨ ¬ 𝜒 ) ) |
4 |
1 2 3
|
3anbi123i |
⊢ ( ( ¬ ( 𝜑 ∧ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜒 ) ∧ ¬ ( 𝜓 ∧ 𝜒 ) ) ↔ ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ∧ ( ¬ 𝜑 ∨ ¬ 𝜒 ) ∧ ( ¬ 𝜓 ∨ ¬ 𝜒 ) ) ) |
5 |
|
3ioran |
⊢ ( ¬ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) ↔ ( ¬ ( 𝜑 ∧ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜒 ) ∧ ¬ ( 𝜓 ∧ 𝜒 ) ) ) |
6 |
|
cador |
⊢ ( cadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) ) |
7 |
5 6
|
xchnxbir |
⊢ ( ¬ cadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ¬ ( 𝜑 ∧ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜒 ) ∧ ¬ ( 𝜓 ∧ 𝜒 ) ) ) |
8 |
|
cadan |
⊢ ( cadd ( ¬ 𝜑 , ¬ 𝜓 , ¬ 𝜒 ) ↔ ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ∧ ( ¬ 𝜑 ∨ ¬ 𝜒 ) ∧ ( ¬ 𝜓 ∨ ¬ 𝜒 ) ) ) |
9 |
4 7 8
|
3bitr4i |
⊢ ( ¬ cadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ cadd ( ¬ 𝜑 , ¬ 𝜓 , ¬ 𝜒 ) ) |