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Mirrors > Home > ILE Home > Th. List > elrabd | GIF version |
Description: Membership in a restricted class abstraction, using implicit substitution. Deduction version of elrab 2835. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
elrabd.1 | ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) |
elrabd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
elrabd.3 | ⊢ (𝜑 → 𝜒) |
Ref | Expression |
---|---|
elrabd | ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrabd.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
2 | elrabd.3 | . . 3 ⊢ (𝜑 → 𝜒) | |
3 | 1, 2 | jca 304 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∧ 𝜒)) |
4 | elrabd.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) | |
5 | 4 | elrab 2835 | . 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓} ↔ (𝐴 ∈ 𝐵 ∧ 𝜒)) |
6 | 3, 5 | sylibr 133 | 1 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ 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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rab 2423 df-v 2683 |
This theorem is referenced by: ctssdccl 6989 suplocexprlemru 7520 suplocexprlemloc 7522 znnen 11900 ennnfonelemj0 11903 ennnfonelemg 11905 cdivcncfap 12745 cnopnap 12752 ivthinc 12779 limcdifap 12789 limcimolemlt 12791 dvcoapbr 12829 subctctexmid 13185 |
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