Step |
Hyp |
Ref |
Expression |
1 |
|
frege116.x |
⊢ 𝑋 ∈ 𝑈 |
2 |
|
dffrege115 |
⊢ ( ∀ 𝑐 ∀ 𝑏 ( 𝑏 𝑅 𝑐 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ) ↔ Fun ◡ ◡ 𝑅 ) |
3 |
1
|
frege68c |
⊢ ( ( ∀ 𝑐 ∀ 𝑏 ( 𝑏 𝑅 𝑐 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ) ↔ Fun ◡ ◡ 𝑅 ) → ( Fun ◡ ◡ 𝑅 → [ 𝑋 / 𝑐 ] ∀ 𝑏 ( 𝑏 𝑅 𝑐 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ) ) ) |
4 |
|
sbcal |
⊢ ( [ 𝑋 / 𝑐 ] ∀ 𝑏 ( 𝑏 𝑅 𝑐 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ) ↔ ∀ 𝑏 [ 𝑋 / 𝑐 ] ( 𝑏 𝑅 𝑐 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ) ) |
5 |
|
sbcimg |
⊢ ( 𝑋 ∈ 𝑈 → ( [ 𝑋 / 𝑐 ] ( 𝑏 𝑅 𝑐 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ) ↔ ( [ 𝑋 / 𝑐 ] 𝑏 𝑅 𝑐 → [ 𝑋 / 𝑐 ] ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ) ) ) |
6 |
1 5
|
ax-mp |
⊢ ( [ 𝑋 / 𝑐 ] ( 𝑏 𝑅 𝑐 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ) ↔ ( [ 𝑋 / 𝑐 ] 𝑏 𝑅 𝑐 → [ 𝑋 / 𝑐 ] ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ) ) |
7 |
|
sbcbr2g |
⊢ ( 𝑋 ∈ 𝑈 → ( [ 𝑋 / 𝑐 ] 𝑏 𝑅 𝑐 ↔ 𝑏 𝑅 ⦋ 𝑋 / 𝑐 ⦌ 𝑐 ) ) |
8 |
1 7
|
ax-mp |
⊢ ( [ 𝑋 / 𝑐 ] 𝑏 𝑅 𝑐 ↔ 𝑏 𝑅 ⦋ 𝑋 / 𝑐 ⦌ 𝑐 ) |
9 |
|
csbvarg |
⊢ ( 𝑋 ∈ 𝑈 → ⦋ 𝑋 / 𝑐 ⦌ 𝑐 = 𝑋 ) |
10 |
1 9
|
ax-mp |
⊢ ⦋ 𝑋 / 𝑐 ⦌ 𝑐 = 𝑋 |
11 |
10
|
breq2i |
⊢ ( 𝑏 𝑅 ⦋ 𝑋 / 𝑐 ⦌ 𝑐 ↔ 𝑏 𝑅 𝑋 ) |
12 |
8 11
|
bitri |
⊢ ( [ 𝑋 / 𝑐 ] 𝑏 𝑅 𝑐 ↔ 𝑏 𝑅 𝑋 ) |
13 |
|
sbcal |
⊢ ( [ 𝑋 / 𝑐 ] ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ↔ ∀ 𝑎 [ 𝑋 / 𝑐 ] ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ) |
14 |
|
sbcimg |
⊢ ( 𝑋 ∈ 𝑈 → ( [ 𝑋 / 𝑐 ] ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ↔ ( [ 𝑋 / 𝑐 ] 𝑏 𝑅 𝑎 → [ 𝑋 / 𝑐 ] 𝑎 = 𝑐 ) ) ) |
15 |
1 14
|
ax-mp |
⊢ ( [ 𝑋 / 𝑐 ] ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ↔ ( [ 𝑋 / 𝑐 ] 𝑏 𝑅 𝑎 → [ 𝑋 / 𝑐 ] 𝑎 = 𝑐 ) ) |
16 |
|
sbcg |
⊢ ( 𝑋 ∈ 𝑈 → ( [ 𝑋 / 𝑐 ] 𝑏 𝑅 𝑎 ↔ 𝑏 𝑅 𝑎 ) ) |
17 |
1 16
|
ax-mp |
⊢ ( [ 𝑋 / 𝑐 ] 𝑏 𝑅 𝑎 ↔ 𝑏 𝑅 𝑎 ) |
18 |
|
sbceq2g |
⊢ ( 𝑋 ∈ 𝑈 → ( [ 𝑋 / 𝑐 ] 𝑎 = 𝑐 ↔ 𝑎 = ⦋ 𝑋 / 𝑐 ⦌ 𝑐 ) ) |
19 |
1 18
|
ax-mp |
⊢ ( [ 𝑋 / 𝑐 ] 𝑎 = 𝑐 ↔ 𝑎 = ⦋ 𝑋 / 𝑐 ⦌ 𝑐 ) |
20 |
10
|
eqeq2i |
⊢ ( 𝑎 = ⦋ 𝑋 / 𝑐 ⦌ 𝑐 ↔ 𝑎 = 𝑋 ) |
21 |
19 20
|
bitri |
⊢ ( [ 𝑋 / 𝑐 ] 𝑎 = 𝑐 ↔ 𝑎 = 𝑋 ) |
22 |
17 21
|
imbi12i |
⊢ ( ( [ 𝑋 / 𝑐 ] 𝑏 𝑅 𝑎 → [ 𝑋 / 𝑐 ] 𝑎 = 𝑐 ) ↔ ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑋 ) ) |
23 |
15 22
|
bitri |
⊢ ( [ 𝑋 / 𝑐 ] ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ↔ ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑋 ) ) |
24 |
23
|
albii |
⊢ ( ∀ 𝑎 [ 𝑋 / 𝑐 ] ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ↔ ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑋 ) ) |
25 |
13 24
|
bitri |
⊢ ( [ 𝑋 / 𝑐 ] ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ↔ ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑋 ) ) |
26 |
12 25
|
imbi12i |
⊢ ( ( [ 𝑋 / 𝑐 ] 𝑏 𝑅 𝑐 → [ 𝑋 / 𝑐 ] ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ) ↔ ( 𝑏 𝑅 𝑋 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑋 ) ) ) |
27 |
6 26
|
bitri |
⊢ ( [ 𝑋 / 𝑐 ] ( 𝑏 𝑅 𝑐 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ) ↔ ( 𝑏 𝑅 𝑋 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑋 ) ) ) |
28 |
27
|
albii |
⊢ ( ∀ 𝑏 [ 𝑋 / 𝑐 ] ( 𝑏 𝑅 𝑐 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ) ↔ ∀ 𝑏 ( 𝑏 𝑅 𝑋 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑋 ) ) ) |
29 |
4 28
|
bitri |
⊢ ( [ 𝑋 / 𝑐 ] ∀ 𝑏 ( 𝑏 𝑅 𝑐 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ) ↔ ∀ 𝑏 ( 𝑏 𝑅 𝑋 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑋 ) ) ) |
30 |
3 29
|
syl6ib |
⊢ ( ( ∀ 𝑐 ∀ 𝑏 ( 𝑏 𝑅 𝑐 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑐 ) ) ↔ Fun ◡ ◡ 𝑅 ) → ( Fun ◡ ◡ 𝑅 → ∀ 𝑏 ( 𝑏 𝑅 𝑋 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑋 ) ) ) ) |
31 |
2 30
|
ax-mp |
⊢ ( Fun ◡ ◡ 𝑅 → ∀ 𝑏 ( 𝑏 𝑅 𝑋 → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → 𝑎 = 𝑋 ) ) ) |