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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 2838 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 2911 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | elrabsf.1 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
3 | nfcv 2281 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
4 | nfv 1508 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
5 | nfsbc1v 2927 | . . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
6 | sbceq1a 2918 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
7 | 2, 3, 4, 5, 6 | cbvrab 2684 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ [𝑦 / 𝑥]𝜑} |
8 | 1, 7 | elrab2 2843 | 1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1480 Ⅎwnfc 2268 {crab 2420 [wsbc 2909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rab 2425 df-v 2688 df-sbc 2910 |
This theorem is referenced by: mpoxopovel 6138 zsupcllemstep 11638 infssuzex 11642 |
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