Step |
Hyp |
Ref |
Expression |
1 |
|
sbbibOLD.y |
⊢ Ⅎ 𝑦 𝜑 |
2 |
|
sbbibOLD.x |
⊢ Ⅎ 𝑥 𝜓 |
3 |
|
nfs1v |
⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 |
4 |
3 2
|
nfbi |
⊢ Ⅎ 𝑥 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) |
5 |
4
|
nf5ri |
⊢ ( ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) → ∀ 𝑥 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) ) |
6 |
5
|
hbal |
⊢ ( ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) → ∀ 𝑥 ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) ) |
7 |
|
sbcov |
⊢ ( [ 𝑥 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜑 ) |
8 |
1
|
sbf |
⊢ ( [ 𝑥 / 𝑦 ] 𝜑 ↔ 𝜑 ) |
9 |
7 8
|
bitri |
⊢ ( [ 𝑥 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 ) |
10 |
|
spsbbi |
⊢ ( ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) → ( [ 𝑥 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜓 ) ) |
11 |
9 10
|
bitr3id |
⊢ ( ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) → ( 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜓 ) ) |
12 |
6 11
|
alrimih |
⊢ ( ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) → ∀ 𝑥 ( 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜓 ) ) |
13 |
|
nfs1v |
⊢ Ⅎ 𝑦 [ 𝑥 / 𝑦 ] 𝜓 |
14 |
1 13
|
nfbi |
⊢ Ⅎ 𝑦 ( 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜓 ) |
15 |
14
|
nf5ri |
⊢ ( ( 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜓 ) → ∀ 𝑦 ( 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜓 ) ) |
16 |
15
|
hbal |
⊢ ( ∀ 𝑥 ( 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜓 ) → ∀ 𝑦 ∀ 𝑥 ( 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜓 ) ) |
17 |
|
spsbbi |
⊢ ( ∀ 𝑥 ( 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜓 ) → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] [ 𝑥 / 𝑦 ] 𝜓 ) ) |
18 |
|
sbcov |
⊢ ( [ 𝑦 / 𝑥 ] [ 𝑥 / 𝑦 ] 𝜓 ↔ [ 𝑦 / 𝑥 ] 𝜓 ) |
19 |
2
|
sbf |
⊢ ( [ 𝑦 / 𝑥 ] 𝜓 ↔ 𝜓 ) |
20 |
18 19
|
bitri |
⊢ ( [ 𝑦 / 𝑥 ] [ 𝑥 / 𝑦 ] 𝜓 ↔ 𝜓 ) |
21 |
17 20
|
bitrdi |
⊢ ( ∀ 𝑥 ( 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜓 ) → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) ) |
22 |
16 21
|
alrimih |
⊢ ( ∀ 𝑥 ( 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜓 ) → ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) ) |
23 |
12 22
|
impbii |
⊢ ( ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) ↔ ∀ 𝑥 ( 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜓 ) ) |