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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 2176 | . 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 2160 {cab 2175 |
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 2176 |
This theorem is referenced by: abeq2 2298 abeq2i 2300 abeq1i 2301 abeq2d 2302 abid2f 2358 elabgt 2893 elabgf 2894 ralab2 2916 rexab2 2918 sbccsbg 3101 sbccsb2g 3102 ss2ab 3238 abn0r 3462 abn0m 3463 tpid3g 3725 eluniab 3839 elintab 3873 iunab 3951 iinab 3966 intexabim 4173 iinexgm 4175 opm 4255 finds2 4621 dmmrnm 4867 iotaexab 5217 sniota 5229 eusvobj2 5886 eloprabga 5987 indpi 7376 4sqlem12 12445 elabgf0 15015 |
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