Step |
Hyp |
Ref |
Expression |
1 |
|
sban |
⊢ ( [ 𝑥 / 𝑏 ] ( 𝜑 ∧ 𝜓 ) ↔ ( [ 𝑥 / 𝑏 ] 𝜑 ∧ [ 𝑥 / 𝑏 ] 𝜓 ) ) |
2 |
1
|
sbbii |
⊢ ( [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] ( 𝜑 ∧ 𝜓 ) ↔ [ 𝑏 / 𝑎 ] ( [ 𝑥 / 𝑏 ] 𝜑 ∧ [ 𝑥 / 𝑏 ] 𝜓 ) ) |
3 |
2
|
sbbii |
⊢ ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] ( 𝜑 ∧ 𝜓 ) ↔ [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] ( [ 𝑥 / 𝑏 ] 𝜑 ∧ [ 𝑥 / 𝑏 ] 𝜓 ) ) |
4 |
|
sban |
⊢ ( [ 𝑏 / 𝑎 ] ( [ 𝑥 / 𝑏 ] 𝜑 ∧ [ 𝑥 / 𝑏 ] 𝜓 ) ↔ ( [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜑 ∧ [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜓 ) ) |
5 |
4
|
sbbii |
⊢ ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] ( [ 𝑥 / 𝑏 ] 𝜑 ∧ [ 𝑥 / 𝑏 ] 𝜓 ) ↔ [ 𝑎 / 𝑥 ] ( [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜑 ∧ [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜓 ) ) |
6 |
|
sban |
⊢ ( [ 𝑎 / 𝑥 ] ( [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜑 ∧ [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜓 ) ↔ ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜑 ∧ [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜓 ) ) |
7 |
3 5 6
|
3bitri |
⊢ ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] ( 𝜑 ∧ 𝜓 ) ↔ ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜑 ∧ [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜓 ) ) |
8 |
|
pm4.38 |
⊢ ( ( ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜑 ↔ 𝜑 ) ∧ ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜓 ↔ 𝜓 ) ) → ( ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜑 ∧ [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜓 ) ↔ ( 𝜑 ∧ 𝜓 ) ) ) |
9 |
7 8
|
syl5bb |
⊢ ( ( ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜑 ↔ 𝜑 ) ∧ ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜓 ↔ 𝜓 ) ) → ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ 𝜓 ) ) ) |
10 |
9
|
alanimi |
⊢ ( ( ∀ 𝑏 ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜑 ↔ 𝜑 ) ∧ ∀ 𝑏 ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜓 ↔ 𝜓 ) ) → ∀ 𝑏 ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ 𝜓 ) ) ) |
11 |
10
|
alanimi |
⊢ ( ( ∀ 𝑎 ∀ 𝑏 ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜑 ↔ 𝜑 ) ∧ ∀ 𝑎 ∀ 𝑏 ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜓 ↔ 𝜓 ) ) → ∀ 𝑎 ∀ 𝑏 ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ 𝜓 ) ) ) |
12 |
|
df-ich |
⊢ ( [ 𝑎 ⇄ 𝑏 ] 𝜑 ↔ ∀ 𝑎 ∀ 𝑏 ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜑 ↔ 𝜑 ) ) |
13 |
|
df-ich |
⊢ ( [ 𝑎 ⇄ 𝑏 ] 𝜓 ↔ ∀ 𝑎 ∀ 𝑏 ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜓 ↔ 𝜓 ) ) |
14 |
12 13
|
anbi12i |
⊢ ( ( [ 𝑎 ⇄ 𝑏 ] 𝜑 ∧ [ 𝑎 ⇄ 𝑏 ] 𝜓 ) ↔ ( ∀ 𝑎 ∀ 𝑏 ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜑 ↔ 𝜑 ) ∧ ∀ 𝑎 ∀ 𝑏 ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] 𝜓 ↔ 𝜓 ) ) ) |
15 |
|
df-ich |
⊢ ( [ 𝑎 ⇄ 𝑏 ] ( 𝜑 ∧ 𝜓 ) ↔ ∀ 𝑎 ∀ 𝑏 ( [ 𝑎 / 𝑥 ] [ 𝑏 / 𝑎 ] [ 𝑥 / 𝑏 ] ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ 𝜓 ) ) ) |
16 |
11 14 15
|
3imtr4i |
⊢ ( ( [ 𝑎 ⇄ 𝑏 ] 𝜑 ∧ [ 𝑎 ⇄ 𝑏 ] 𝜓 ) → [ 𝑎 ⇄ 𝑏 ] ( 𝜑 ∧ 𝜓 ) ) |