Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme31snd.d |
⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
2 |
|
cdleme31snd.n |
⊢ 𝑁 = ( ( 𝑣 ∨ 𝑉 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝑣 ) ∧ 𝑊 ) ) ) |
3 |
|
cdleme31snd.e |
⊢ 𝐸 = ( ( 𝑂 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑂 ) ∧ 𝑊 ) ) ) |
4 |
|
cdleme31snd.o |
⊢ 𝑂 = ( ( 𝑆 ∨ 𝑉 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝑆 ) ∧ 𝑊 ) ) ) |
5 |
|
csbnestgw |
⊢ ( 𝑆 ∈ 𝐴 → ⦋ 𝑆 / 𝑣 ⦌ ⦋ 𝑁 / 𝑡 ⦌ 𝐷 = ⦋ ⦋ 𝑆 / 𝑣 ⦌ 𝑁 / 𝑡 ⦌ 𝐷 ) |
6 |
2 4
|
cdleme31sc |
⊢ ( 𝑆 ∈ 𝐴 → ⦋ 𝑆 / 𝑣 ⦌ 𝑁 = 𝑂 ) |
7 |
6
|
csbeq1d |
⊢ ( 𝑆 ∈ 𝐴 → ⦋ ⦋ 𝑆 / 𝑣 ⦌ 𝑁 / 𝑡 ⦌ 𝐷 = ⦋ 𝑂 / 𝑡 ⦌ 𝐷 ) |
8 |
4
|
ovexi |
⊢ 𝑂 ∈ V |
9 |
1 3
|
cdleme31sc |
⊢ ( 𝑂 ∈ V → ⦋ 𝑂 / 𝑡 ⦌ 𝐷 = 𝐸 ) |
10 |
8 9
|
ax-mp |
⊢ ⦋ 𝑂 / 𝑡 ⦌ 𝐷 = 𝐸 |
11 |
7 10
|
eqtrdi |
⊢ ( 𝑆 ∈ 𝐴 → ⦋ ⦋ 𝑆 / 𝑣 ⦌ 𝑁 / 𝑡 ⦌ 𝐷 = 𝐸 ) |
12 |
5 11
|
eqtrd |
⊢ ( 𝑆 ∈ 𝐴 → ⦋ 𝑆 / 𝑣 ⦌ ⦋ 𝑁 / 𝑡 ⦌ 𝐷 = 𝐸 ) |