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Mirrors > Home > ILE Home > Th. List > rabeq2i | GIF version |
Description: Inference from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.) |
Ref | Expression |
---|---|
rabeq2i.1 | ⊢ 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝜑} |
Ref | Expression |
---|---|
rabeq2i | ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeq2i.1 | . . 3 ⊢ 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝜑} | |
2 | 1 | eleq2i 2166 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑}) |
3 | rabid 2564 | . 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)) | |
4 | 2, 3 | bitri 183 | 1 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1299 ∈ 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: tfis 4435 fvmptssdm 5437 |
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