Step |
Hyp |
Ref |
Expression |
1 |
|
sbn |
⊢ ( [ 𝑦 / 𝑥 ] ¬ ( 𝜑 → ¬ 𝜓 ) ↔ ¬ [ 𝑦 / 𝑥 ] ( 𝜑 → ¬ 𝜓 ) ) |
2 |
|
sbim |
⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → ¬ 𝜓 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] ¬ 𝜓 ) ) |
3 |
|
sbn |
⊢ ( [ 𝑦 / 𝑥 ] ¬ 𝜓 ↔ ¬ [ 𝑦 / 𝑥 ] 𝜓 ) |
4 |
3
|
imbi2i |
⊢ ( ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] ¬ 𝜓 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 → ¬ [ 𝑦 / 𝑥 ] 𝜓 ) ) |
5 |
2 4
|
bitri |
⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → ¬ 𝜓 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 → ¬ [ 𝑦 / 𝑥 ] 𝜓 ) ) |
6 |
1 5
|
xchbinx |
⊢ ( [ 𝑦 / 𝑥 ] ¬ ( 𝜑 → ¬ 𝜓 ) ↔ ¬ ( [ 𝑦 / 𝑥 ] 𝜑 → ¬ [ 𝑦 / 𝑥 ] 𝜓 ) ) |
7 |
|
df-an |
⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ¬ ( 𝜑 → ¬ 𝜓 ) ) |
8 |
7
|
sbbii |
⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 ∧ 𝜓 ) ↔ [ 𝑦 / 𝑥 ] ¬ ( 𝜑 → ¬ 𝜓 ) ) |
9 |
|
df-an |
⊢ ( ( [ 𝑦 / 𝑥 ] 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜓 ) ↔ ¬ ( [ 𝑦 / 𝑥 ] 𝜑 → ¬ [ 𝑦 / 𝑥 ] 𝜓 ) ) |
10 |
6 8 9
|
3bitr4i |
⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 ∧ 𝜓 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜓 ) ) |