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| Mirrors > Home > MPE Home > Th. List > elrabsf | Structured version Visualization version GIF version | ||
| Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3688 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 3790 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
| 2 | elrabsf.1 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
| 3 | nfcv 2905 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
| 4 | nfv 1914 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
| 5 | nfsbc1v 3808 | . . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
| 6 | sbceq1a 3799 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
| 7 | 2, 3, 4, 5, 6 | cbvrabw 3473 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ [𝑦 / 𝑥]𝜑} |
| 8 | 1, 7 | elrab2 3695 | 1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 Ⅎwnfc 2890 {crab 3436 [wsbc 3788 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-sbc 3789 |
| This theorem is referenced by: frpoinsg 6364 wfisgOLD 6375 onminesb 7813 tfisg 7875 mpoxopovel 8245 frinsg 9791 ac6num 10519 hashrabsn1 14413 bnj23 34732 bnj1204 35026 weiunlem2 36464 rabrenfdioph 42825 |
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