Metamath Proof Explorer
Description: Class element of a restricted class abstraction. (Contributed by Peter
Mazsa, 24-Jul-2021)
|
|
Ref |
Expression |
|
Hypotheses |
rabeqel.1 |
⊢ 𝐵 = { 𝑥 ∈ 𝐴 ∣ 𝜑 } |
|
|
rabeqel.2 |
⊢ ( 𝑥 = 𝐶 → ( 𝜑 ↔ 𝜓 ) ) |
|
Assertion |
rabeqel |
⊢ ( 𝐶 ∈ 𝐵 ↔ ( 𝜓 ∧ 𝐶 ∈ 𝐴 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
rabeqel.1 |
⊢ 𝐵 = { 𝑥 ∈ 𝐴 ∣ 𝜑 } |
2 |
|
rabeqel.2 |
⊢ ( 𝑥 = 𝐶 → ( 𝜑 ↔ 𝜓 ) ) |
3 |
2 1
|
elrab2 |
⊢ ( 𝐶 ∈ 𝐵 ↔ ( 𝐶 ∈ 𝐴 ∧ 𝜓 ) ) |
4 |
3
|
biancomi |
⊢ ( 𝐶 ∈ 𝐵 ↔ ( 𝜓 ∧ 𝐶 ∈ 𝐴 ) ) |