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Mirrors > Home > ILE Home > Th. List > elrabsf | GIF version |
Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2696 has implicit substitution). The hypothesis specifies that 𝑥 must not be a free variable in 𝐵. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
elrabsf.1 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
elrabsf | ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq 2766 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | elrabsf.1 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
3 | nfcv 2178 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
4 | nfv 1421 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
5 | nfsbc1v 2782 | . . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
6 | sbceq1a 2773 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
7 | 2, 3, 4, 5, 6 | cbvrab 2555 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ [𝑦 / 𝑥]𝜑} |
8 | 1, 7 | elrab2 2700 | 1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∈ wcel 1393 Ⅎwnfc 2165 {crab 2310 [wsbc 2764 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rab 2315 df-v 2559 df-sbc 2765 |
This theorem is referenced by: mpt2xopovel 5856 |
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