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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 2204 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑}) |
3 | rabid 2604 | . 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 1331 ∈ wcel 1480 {crab 2418 |
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-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-rab 2423 |
This theorem is referenced by: tfis 4492 fvmptssdm 5498 suplocsrlempr 7608 suplocsrlem 7609 |
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