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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 1508 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1508 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvabv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvab 2263 | 1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 {cab 2125 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 |
This theorem is referenced by: cdeqab1 2901 difjust 3072 unjust 3074 injust 3076 uniiunlem 3185 dfif3 3487 pwjust 3511 snjust 3532 intab 3800 iotajust 5087 tfrlemi1 6229 tfr1onlemaccex 6245 tfrcllemaccex 6258 frecsuc 6304 isbth 6855 nqprlu 7355 recexpr 7446 caucvgprprlemval 7496 caucvgprprlemnbj 7501 caucvgprprlemaddq 7516 caucvgprprlem1 7517 caucvgprprlem2 7518 axcaucvg 7708 mertensabs 11306 bds 13049 |
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