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Mirrors > Home > MPE Home > Th. List > elrab | Structured version Visualization version GIF version |
Description: Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.) Remove dependency on ax-13 2373. (Revised by Steven Nguyen, 23-Nov-2022.) |
Ref | Expression |
---|---|
elrab.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
elrab | ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3418 | . 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} → 𝐴 ∈ V) | |
2 | elex 3418 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | 2 | adantr 484 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → 𝐴 ∈ V) |
4 | eleq1 2821 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
5 | elrab.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | 4, 5 | anbi12d 634 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))) |
7 | df-rab 3063 | . . 3 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} | |
8 | 6, 7 | elab2g 3580 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))) |
9 | 1, 3, 8 | pm5.21nii 383 | 1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
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