Step |
Hyp |
Ref |
Expression |
1 |
|
ichcircshi.1 |
⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜓 ) ) |
2 |
|
ichcircshi.2 |
⊢ ( 𝑦 = 𝑥 → ( 𝜓 ↔ 𝜒 ) ) |
3 |
|
ichcircshi.3 |
⊢ ( 𝑧 = 𝑦 → ( 𝜒 ↔ 𝜑 ) ) |
4 |
3
|
bicomd |
⊢ ( 𝑧 = 𝑦 → ( 𝜑 ↔ 𝜒 ) ) |
5 |
4
|
equcoms |
⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜒 ) ) |
6 |
5
|
sbievw |
⊢ ( [ 𝑧 / 𝑦 ] 𝜑 ↔ 𝜒 ) |
7 |
6
|
2sbbii |
⊢ ( [ 𝑥 / 𝑧 ] [ 𝑦 / 𝑥 ] [ 𝑧 / 𝑦 ] 𝜑 ↔ [ 𝑥 / 𝑧 ] [ 𝑦 / 𝑥 ] 𝜒 ) |
8 |
2
|
bicomd |
⊢ ( 𝑦 = 𝑥 → ( 𝜒 ↔ 𝜓 ) ) |
9 |
8
|
equcoms |
⊢ ( 𝑥 = 𝑦 → ( 𝜒 ↔ 𝜓 ) ) |
10 |
9
|
sbievw |
⊢ ( [ 𝑦 / 𝑥 ] 𝜒 ↔ 𝜓 ) |
11 |
10
|
sbbii |
⊢ ( [ 𝑥 / 𝑧 ] [ 𝑦 / 𝑥 ] 𝜒 ↔ [ 𝑥 / 𝑧 ] 𝜓 ) |
12 |
1
|
bicomd |
⊢ ( 𝑥 = 𝑧 → ( 𝜓 ↔ 𝜑 ) ) |
13 |
12
|
equcoms |
⊢ ( 𝑧 = 𝑥 → ( 𝜓 ↔ 𝜑 ) ) |
14 |
13
|
sbievw |
⊢ ( [ 𝑥 / 𝑧 ] 𝜓 ↔ 𝜑 ) |
15 |
7 11 14
|
3bitri |
⊢ ( [ 𝑥 / 𝑧 ] [ 𝑦 / 𝑥 ] [ 𝑧 / 𝑦 ] 𝜑 ↔ 𝜑 ) |
16 |
15
|
gen2 |
⊢ ∀ 𝑥 ∀ 𝑦 ( [ 𝑥 / 𝑧 ] [ 𝑦 / 𝑥 ] [ 𝑧 / 𝑦 ] 𝜑 ↔ 𝜑 ) |
17 |
|
df-ich |
⊢ ( [ 𝑥 ⇄ 𝑦 ] 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( [ 𝑥 / 𝑧 ] [ 𝑦 / 𝑥 ] [ 𝑧 / 𝑦 ] 𝜑 ↔ 𝜑 ) ) |
18 |
16 17
|
mpbir |
⊢ [ 𝑥 ⇄ 𝑦 ] 𝜑 |