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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 2481 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | 1 | abeq2i 2304 | 1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∈ wcel 2164 {crab 2476 |
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-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-rab 2481 |
This theorem is referenced by: rabeq2i 2757 rabn0m 3474 repizf2lem 4190 rabxfrd 4498 onintrab2im 4546 tfis 4611 nnwosdc 12163 imasnopn 14438 |
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