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Mirrors > Home > ILE Home > Th. List > abid2 | GIF version |
Description: A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.) |
Ref | Expression |
---|---|
abid2 | ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biid 170 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) | |
2 | 1 | abbi2i 2252 | . 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} |
3 | 2 | eqcomi 2141 | 1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 {cab 2123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 |
This theorem is referenced by: csbid 3006 abss 3161 ssab 3162 abssi 3167 notab 3341 inrab2 3344 dfrab2 3346 dfrab3 3347 notrab 3348 eusn 3592 dfopg 3698 iunid 3863 csbexga 4051 imai 4890 dffv4g 5411 frec0g 6287 dfixp 6587 euen1b 6690 acfun 7056 ccfunen 7072 |
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