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| Description: Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.) (Proof shortened by BJ, 22-Apr-2019.) | 
| Ref | Expression | 
|---|---|
| dfrab2 | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfrab3 4318 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑}) | |
| 2 | incom 4208 | . 2 ⊢ (𝐴 ∩ {𝑥 ∣ 𝜑}) = ({𝑥 ∣ 𝜑} ∩ 𝐴) | |
| 3 | 1, 2 | eqtri 2764 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 {cab 2713 {crab 3435 ∩ cin 3949 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-in 3957 | 
| This theorem is referenced by: rabdif 4320 dfpred3 6331 predres 6359 lubdm 18397 glbdm 18410 psrbagsn 22088 ismbl 25562 lrrecse 27976 lrrecpred 27978 eulerpartgbij 34375 orvcval4 34464 fvline2 36148 abeqin 38254 nznngen 44340 | 
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