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Mirrors > Home > ILE Home > Th. List > abid | GIF version |
Description: Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
abid | ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clab 2180 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ [𝑥 / 𝑥]𝜑) | |
2 | sbid 1785 | . 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | |
3 | 1, 2 | bitri 184 | 1 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 [wsb 1773 ∈ wcel 2164 {cab 2179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-4 1521 ax-17 1537 ax-i9 1541 |
This theorem depends on definitions: df-bi 117 df-sb 1774 df-clab 2180 |
This theorem is referenced by: abeq2 2302 abeq2i 2304 abeq1i 2305 abeq2d 2306 abid2f 2362 elabgt 2902 elabgf 2903 ralab2 2925 rexab2 2927 sbccsbg 3110 sbccsb2g 3111 ss2ab 3248 abn0r 3472 abn0m 3473 tpid3g 3734 eluniab 3848 elintab 3882 iunab 3960 iinab 3975 intexabim 4182 iinexgm 4184 opm 4264 finds2 4634 dmmrnm 4882 iotaexab 5234 sniota 5246 eusvobj2 5905 eloprabga 6006 indpi 7404 4sqlem12 12543 elabgf0 15339 |
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