Step |
Hyp |
Ref |
Expression |
1 |
|
dfcleq |
⊢ ( { 𝑥 ∣ 𝜑 } = { 𝑥 ∣ 𝜓 } ↔ ∀ 𝑦 ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝑦 ∈ { 𝑥 ∣ 𝜓 } ) ) |
2 |
|
nfsab1 |
⊢ Ⅎ 𝑥 𝑦 ∈ { 𝑥 ∣ 𝜑 } |
3 |
|
nfsab1 |
⊢ Ⅎ 𝑥 𝑦 ∈ { 𝑥 ∣ 𝜓 } |
4 |
2 3
|
nfbi |
⊢ Ⅎ 𝑥 ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝑦 ∈ { 𝑥 ∣ 𝜓 } ) |
5 |
|
nfv |
⊢ Ⅎ 𝑦 ( 𝜑 ↔ 𝜓 ) |
6 |
|
df-clab |
⊢ ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } ↔ [ 𝑦 / 𝑥 ] 𝜑 ) |
7 |
|
sbequ12r |
⊢ ( 𝑦 = 𝑥 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 ) ) |
8 |
6 7
|
bitrid |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜑 ) ) |
9 |
|
df-clab |
⊢ ( 𝑦 ∈ { 𝑥 ∣ 𝜓 } ↔ [ 𝑦 / 𝑥 ] 𝜓 ) |
10 |
|
sbequ12r |
⊢ ( 𝑦 = 𝑥 → ( [ 𝑦 / 𝑥 ] 𝜓 ↔ 𝜓 ) ) |
11 |
9 10
|
bitrid |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ { 𝑥 ∣ 𝜓 } ↔ 𝜓 ) ) |
12 |
8 11
|
bibi12d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝑦 ∈ { 𝑥 ∣ 𝜓 } ) ↔ ( 𝜑 ↔ 𝜓 ) ) ) |
13 |
4 5 12
|
cbvalv1 |
⊢ ( ∀ 𝑦 ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝑦 ∈ { 𝑥 ∣ 𝜓 } ) ↔ ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) ) |
14 |
1 13
|
bitr2i |
⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) ↔ { 𝑥 ∣ 𝜑 } = { 𝑥 ∣ 𝜓 } ) |