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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 171 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) | |
2 | 1 | abbi2i 2304 | . 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} |
3 | 2 | eqcomi 2193 | 1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ 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-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 |
This theorem is referenced by: csbid 3080 abss 3239 ssab 3240 abssi 3245 notab 3420 inrab2 3423 dfrab2 3425 dfrab3 3426 notrab 3427 eusn 3681 dfopg 3791 iunid 3957 csbexga 4146 imai 5002 dffv4g 5531 frec0g 6423 dfixp 6727 euen1b 6830 acfun 7237 ccfunen 7294 |
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