Step |
Hyp |
Ref |
Expression |
1 |
|
frege70.x |
⊢ 𝑋 ∈ 𝑉 |
2 |
|
dffrege69 |
⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) ↔ 𝑅 hereditary 𝐴 ) |
3 |
1
|
frege68c |
⊢ ( ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) ↔ 𝑅 hereditary 𝐴 ) → ( 𝑅 hereditary 𝐴 → [ 𝑋 / 𝑥 ] ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) ) ) |
4 |
|
sbcel1v |
⊢ ( [ 𝑋 / 𝑥 ] 𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴 ) |
5 |
4
|
biimpri |
⊢ ( 𝑋 ∈ 𝐴 → [ 𝑋 / 𝑥 ] 𝑥 ∈ 𝐴 ) |
6 |
|
sbcim1 |
⊢ ( [ 𝑋 / 𝑥 ] ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) → ( [ 𝑋 / 𝑥 ] 𝑥 ∈ 𝐴 → [ 𝑋 / 𝑥 ] ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) ) |
7 |
|
sbcal |
⊢ ( [ 𝑋 / 𝑥 ] ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ↔ ∀ 𝑦 [ 𝑋 / 𝑥 ] ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) |
8 |
|
sbcim1 |
⊢ ( [ 𝑋 / 𝑥 ] ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) → ( [ 𝑋 / 𝑥 ] 𝑥 𝑅 𝑦 → [ 𝑋 / 𝑥 ] 𝑦 ∈ 𝐴 ) ) |
9 |
|
sbcbr1g |
⊢ ( 𝑋 ∈ 𝑉 → ( [ 𝑋 / 𝑥 ] 𝑥 𝑅 𝑦 ↔ ⦋ 𝑋 / 𝑥 ⦌ 𝑥 𝑅 𝑦 ) ) |
10 |
1 9
|
ax-mp |
⊢ ( [ 𝑋 / 𝑥 ] 𝑥 𝑅 𝑦 ↔ ⦋ 𝑋 / 𝑥 ⦌ 𝑥 𝑅 𝑦 ) |
11 |
|
csbvarg |
⊢ ( 𝑋 ∈ 𝑉 → ⦋ 𝑋 / 𝑥 ⦌ 𝑥 = 𝑋 ) |
12 |
1 11
|
ax-mp |
⊢ ⦋ 𝑋 / 𝑥 ⦌ 𝑥 = 𝑋 |
13 |
12
|
breq1i |
⊢ ( ⦋ 𝑋 / 𝑥 ⦌ 𝑥 𝑅 𝑦 ↔ 𝑋 𝑅 𝑦 ) |
14 |
10 13
|
bitri |
⊢ ( [ 𝑋 / 𝑥 ] 𝑥 𝑅 𝑦 ↔ 𝑋 𝑅 𝑦 ) |
15 |
|
sbcg |
⊢ ( 𝑋 ∈ 𝑉 → ( [ 𝑋 / 𝑥 ] 𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) ) |
16 |
1 15
|
ax-mp |
⊢ ( [ 𝑋 / 𝑥 ] 𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) |
17 |
8 14 16
|
3imtr3g |
⊢ ( [ 𝑋 / 𝑥 ] ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) → ( 𝑋 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) |
18 |
17
|
alimi |
⊢ ( ∀ 𝑦 [ 𝑋 / 𝑥 ] ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) → ∀ 𝑦 ( 𝑋 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) |
19 |
7 18
|
sylbi |
⊢ ( [ 𝑋 / 𝑥 ] ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) → ∀ 𝑦 ( 𝑋 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) |
20 |
5 6 19
|
syl56 |
⊢ ( [ 𝑋 / 𝑥 ] ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) → ( 𝑋 ∈ 𝐴 → ∀ 𝑦 ( 𝑋 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) ) |
21 |
3 20
|
syl6 |
⊢ ( ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑥 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) ↔ 𝑅 hereditary 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝑋 ∈ 𝐴 → ∀ 𝑦 ( 𝑋 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) ) ) |
22 |
2 21
|
ax-mp |
⊢ ( 𝑅 hereditary 𝐴 → ( 𝑋 ∈ 𝐴 → ∀ 𝑦 ( 𝑋 𝑅 𝑦 → 𝑦 ∈ 𝐴 ) ) ) |