Description: Inference from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | reqabi.1 | ⊢ 𝐴 = { 𝑥 ∈ 𝐵 ∣ 𝜑 } | |
| Assertion | reqabi | ⊢ ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | reqabi.1 | ⊢ 𝐴 = { 𝑥 ∈ 𝐵 ∣ 𝜑 } | |
| 2 | 1 | eleq2i | ⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ) |
| 3 | rabid | ⊢ ( 𝑥 ∈ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) | |
| 4 | 2 3 | bitri | ⊢ ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) |