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Mirrors > Home > MPE Home > Th. List > dfrab2 | Structured version Visualization version GIF version |
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 4276 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑}) | |
2 | incom 4176 | . 2 ⊢ (𝐴 ∩ {𝑥 ∣ 𝜑}) = ({𝑥 ∣ 𝜑} ∩ 𝐴) | |
3 | 1, 2 | eqtri 2842 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1531 {cab 2797 {crab 3140 ∩ cin 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-rab 3145 df-v 3495 df-in 3941 |
This theorem is referenced by: dfpred3 6151 lubdm 17581 glbdm 17594 psrbagsn 20267 ismbl 24119 eulerpartgbij 31623 orvcval4 31711 fvline2 33600 abeqin 35506 rabdif 39098 nznngen 40639 |
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