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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 2308 | . 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} |
3 | 2 | eqcomi 2197 | 1 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ 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-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 2175 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 |
This theorem is referenced by: csbid 3088 abss 3248 ssab 3249 abssi 3254 notab 3429 inrab2 3432 dfrab2 3434 dfrab3 3435 notrab 3436 eusn 3692 dfopg 3802 iunid 3968 csbexga 4157 imai 5021 dffv4g 5551 frec0g 6450 dfixp 6754 euen1b 6857 acfun 7267 ccfunen 7324 |
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