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Mirrors > Home > ILE Home > Th. List > rabid | GIF version |
Description: An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.) |
Ref | Expression |
---|---|
rabid | ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 2384 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | 1 | abeq2i 2210 | 1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1448 {crab 2379 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1391 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-rab 2384 |
This theorem is referenced by: rabeq2i 2638 rabn0m 3337 repizf2lem 4025 rabxfrd 4328 onintrab2im 4372 tfis 4435 imasnopn 12249 |
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