Step |
Hyp |
Ref |
Expression |
1 |
|
ceqsrex2v.1 |
⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) |
2 |
|
ceqsrex2v.2 |
⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) ) |
3 |
|
anass |
⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ 𝜑 ) ↔ ( 𝑥 = 𝐴 ∧ ( 𝑦 = 𝐵 ∧ 𝜑 ) ) ) |
4 |
3
|
rexbii |
⊢ ( ∃ 𝑦 ∈ 𝐷 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ 𝜑 ) ↔ ∃ 𝑦 ∈ 𝐷 ( 𝑥 = 𝐴 ∧ ( 𝑦 = 𝐵 ∧ 𝜑 ) ) ) |
5 |
|
r19.42v |
⊢ ( ∃ 𝑦 ∈ 𝐷 ( 𝑥 = 𝐴 ∧ ( 𝑦 = 𝐵 ∧ 𝜑 ) ) ↔ ( 𝑥 = 𝐴 ∧ ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜑 ) ) ) |
6 |
4 5
|
bitri |
⊢ ( ∃ 𝑦 ∈ 𝐷 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ 𝜑 ) ↔ ( 𝑥 = 𝐴 ∧ ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜑 ) ) ) |
7 |
6
|
rexbii |
⊢ ( ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ∈ 𝐷 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ 𝜑 ) ↔ ∃ 𝑥 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜑 ) ) ) |
8 |
1
|
anbi2d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑦 = 𝐵 ∧ 𝜑 ) ↔ ( 𝑦 = 𝐵 ∧ 𝜓 ) ) ) |
9 |
8
|
rexbidv |
⊢ ( 𝑥 = 𝐴 → ( ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜑 ) ↔ ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜓 ) ) ) |
10 |
9
|
ceqsrexv |
⊢ ( 𝐴 ∈ 𝐶 → ( ∃ 𝑥 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜑 ) ) ↔ ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜓 ) ) ) |
11 |
7 10
|
bitrid |
⊢ ( 𝐴 ∈ 𝐶 → ( ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ∈ 𝐷 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ 𝜑 ) ↔ ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜓 ) ) ) |
12 |
2
|
ceqsrexv |
⊢ ( 𝐵 ∈ 𝐷 → ( ∃ 𝑦 ∈ 𝐷 ( 𝑦 = 𝐵 ∧ 𝜓 ) ↔ 𝜒 ) ) |
13 |
11 12
|
sylan9bb |
⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( ∃ 𝑥 ∈ 𝐶 ∃ 𝑦 ∈ 𝐷 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ 𝜑 ) ↔ 𝜒 ) ) |