Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( 𝜑 ↔ 𝜓 ) ) |
2 |
1
|
imbi1d |
⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( ( 𝜑 → 𝜏 ) ↔ ( 𝜓 → 𝜏 ) ) ) |
3 |
|
notbi |
⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ¬ 𝜑 ↔ ¬ 𝜓 ) ) |
4 |
|
imbi12 |
⊢ ( ( ¬ 𝜑 ↔ ¬ 𝜓 ) → ( ( 𝜒 ↔ 𝜃 ) → ( ( ¬ 𝜑 → 𝜒 ) ↔ ( ¬ 𝜓 → 𝜃 ) ) ) ) |
5 |
3 4
|
sylbi |
⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜒 ↔ 𝜃 ) → ( ( ¬ 𝜑 → 𝜒 ) ↔ ( ¬ 𝜓 → 𝜃 ) ) ) ) |
6 |
5
|
imp |
⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( ( ¬ 𝜑 → 𝜒 ) ↔ ( ¬ 𝜓 → 𝜃 ) ) ) |
7 |
2 6
|
anbi12d |
⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( ( ( 𝜑 → 𝜏 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) ↔ ( ( 𝜓 → 𝜏 ) ∧ ( ¬ 𝜓 → 𝜃 ) ) ) ) |
8 |
|
dfifp2 |
⊢ ( if- ( 𝜑 , 𝜏 , 𝜒 ) ↔ ( ( 𝜑 → 𝜏 ) ∧ ( ¬ 𝜑 → 𝜒 ) ) ) |
9 |
|
dfifp2 |
⊢ ( if- ( 𝜓 , 𝜏 , 𝜃 ) ↔ ( ( 𝜓 → 𝜏 ) ∧ ( ¬ 𝜓 → 𝜃 ) ) ) |
10 |
7 8 9
|
3bitr4g |
⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ( 𝜒 ↔ 𝜃 ) ) → ( if- ( 𝜑 , 𝜏 , 𝜒 ) ↔ if- ( 𝜓 , 𝜏 , 𝜃 ) ) ) |