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Mirrors > Home > ILE Home > Th. List > cbvrabv | GIF version |
Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.) |
Ref | Expression |
---|---|
cbvrabv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvrabv | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2282 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2282 | . 2 ⊢ Ⅎ𝑦𝐴 | |
3 | nfv 1509 | . 2 ⊢ Ⅎ𝑦𝜑 | |
4 | nfv 1509 | . 2 ⊢ Ⅎ𝑥𝜓 | |
5 | cbvrabv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 1, 2, 3, 4, 5 | cbvrab 2687 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 {crab 2421 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rab 2426 |
This theorem is referenced by: pwnss 4091 acexmidlemv 5780 exmidac 7082 genipv 7341 ltexpri 7445 suplocsrlempr 7639 suplocsr 7641 sqne2sq 11891 |
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