Step |
Hyp |
Ref |
Expression |
1 |
|
sbcan |
⊢ ( [ 𝐴 / 𝑥 ] ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ( [ 𝐴 / 𝑥 ] ( 𝜑 ∧ 𝜓 ) ∧ [ 𝐴 / 𝑥 ] 𝜒 ) ) |
2 |
|
sbcan |
⊢ ( [ 𝐴 / 𝑥 ] ( 𝜑 ∧ 𝜓 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 ∧ [ 𝐴 / 𝑥 ] 𝜓 ) ) |
3 |
1 2
|
bianbi |
⊢ ( [ 𝐴 / 𝑥 ] ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ( ( [ 𝐴 / 𝑥 ] 𝜑 ∧ [ 𝐴 / 𝑥 ] 𝜓 ) ∧ [ 𝐴 / 𝑥 ] 𝜒 ) ) |
4 |
|
df-3an |
⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ) |
5 |
4
|
sbcbii |
⊢ ( [ 𝐴 / 𝑥 ] ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ [ 𝐴 / 𝑥 ] ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ) |
6 |
|
df-3an |
⊢ ( ( [ 𝐴 / 𝑥 ] 𝜑 ∧ [ 𝐴 / 𝑥 ] 𝜓 ∧ [ 𝐴 / 𝑥 ] 𝜒 ) ↔ ( ( [ 𝐴 / 𝑥 ] 𝜑 ∧ [ 𝐴 / 𝑥 ] 𝜓 ) ∧ [ 𝐴 / 𝑥 ] 𝜒 ) ) |
7 |
3 5 6
|
3bitr4i |
⊢ ( [ 𝐴 / 𝑥 ] ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 ∧ [ 𝐴 / 𝑥 ] 𝜓 ∧ [ 𝐴 / 𝑥 ] 𝜒 ) ) |