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| Mirrors > Home > MPE Home > Th. List > elrab2 | Structured version Visualization version GIF version | ||
| Description: Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.) |
| Ref | Expression |
|---|---|
| elrab2.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
| elrab2.2 | ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ 𝜑} |
| Ref | Expression |
|---|---|
| elrab2 | ⊢ (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elrab2.2 | . . 3 ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ 𝜑} | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑}) |
| 3 | elrab2.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 4 | 3 | elrab 3692 | . 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
| 5 | 2, 4 | bitri 275 | 1 ⊢ (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
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