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Mirrors > Home > ILE Home > Th. List > cbvprodi | GIF version |
Description: Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Ref | Expression |
---|---|
cbvprodi.1 | ⊢ Ⅎ𝑘𝐵 |
cbvprodi.2 | ⊢ Ⅎ𝑗𝐶 |
cbvprodi.3 | ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbvprodi | ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvprodi.3 | . 2 ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶) | |
2 | nfcv 2281 | . 2 ⊢ Ⅎ𝑘𝐴 | |
3 | nfcv 2281 | . 2 ⊢ Ⅎ𝑗𝐴 | |
4 | cbvprodi.1 | . 2 ⊢ Ⅎ𝑘𝐵 | |
5 | cbvprodi.2 | . 2 ⊢ Ⅎ𝑗𝐶 | |
6 | 1, 2, 3, 4, 5 | cbvprod 11332 | 1 ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶 |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 Ⅎwnfc 2268 ∏cprod 11324 |
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-in1 603 ax-in2 604 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-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-recs 6202 df-frec 6288 df-seqfrec 10224 df-proddc 11325 |
This theorem is referenced by: (None) |
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