Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2254 | . 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} |
3 | 2 | eqcomi 2143 | 1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 {cab 2125 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 |
This theorem is referenced by: csbid 3011 abss 3166 ssab 3167 abssi 3172 notab 3346 inrab2 3349 dfrab2 3351 dfrab3 3352 notrab 3353 eusn 3597 dfopg 3703 iunid 3868 csbexga 4056 imai 4895 dffv4g 5418 frec0g 6294 dfixp 6594 euen1b 6697 acfun 7063 ccfunen 7079 |
Copyright terms: Public domain | W3C validator |