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Mirrors > Home > ILE Home > Th. List > cbvabv | GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.) |
Ref | Expression |
---|---|
cbvabv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvabv | ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1421 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1421 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvabv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvab 2160 | 1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 {cab 2026 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 |
This theorem is referenced by: cdeqab1 2756 difjust 2919 unjust 2921 injust 2923 uniiunlem 3028 dfif3 3343 pwjust 3360 snjust 3380 intab 3644 iotajust 4866 tfrlemi1 5946 frecsuc 5991 nqprlu 6645 recexpr 6736 caucvgprprlemval 6786 caucvgprprlemnbj 6791 caucvgprprlemaddq 6806 caucvgprprlem1 6807 caucvgprprlem2 6808 axcaucvg 6974 bds 9971 |
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