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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 2219 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2219 | . 2 ⊢ Ⅎ𝑦𝐴 | |
3 | nfv 1461 | . 2 ⊢ Ⅎ𝑦𝜑 | |
4 | nfv 1461 | . 2 ⊢ Ⅎ𝑥𝜓 | |
5 | cbvrabv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 1, 2, 3, 4, 5 | cbvrab 2599 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 {crab 2352 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rab 2357 |
This theorem is referenced by: pwnss 3933 acexmidlemv 5530 genipv 6699 ltexpri 6803 sqne2sq 10555 |
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