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| Mirrors > Home > MPE Home > Th. List > elrabd | Structured version Visualization version GIF version | ||
| Description: Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 3692. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| Ref | Expression |
|---|---|
| elrabd.1 | ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) |
| elrabd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
| elrabd.3 | ⊢ (𝜑 → 𝜒) |
| Ref | Expression |
|---|---|
| elrabd | ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elrabd.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
| 2 | elrabd.3 | . 2 ⊢ (𝜑 → 𝜒) | |
| 3 | elrabd.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) | |
| 4 | 3 | elrab 3692 | . 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓} ↔ (𝐴 ∈ 𝐵 ∧ 𝜒)) |
| 5 | 1, 2, 4 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓}) |
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