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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 2319 | . 2 ⊢ Ⅎ𝑘𝐴 | |
3 | nfcv 2319 | . 2 ⊢ Ⅎ𝑗𝐴 | |
4 | cbvprodi.1 | . 2 ⊢ Ⅎ𝑘𝐵 | |
5 | cbvprodi.2 | . 2 ⊢ Ⅎ𝑗𝐶 | |
6 | 1, 2, 3, 4, 5 | cbvprod 11565 | 1 ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶 |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 Ⅎwnfc 2306 ∏cprod 11557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-if 3535 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-cnv 4634 df-dm 4636 df-rn 4637 df-res 4638 df-iota 5178 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-recs 6305 df-frec 6391 df-seqfrec 10445 df-proddc 11558 |
This theorem is referenced by: prodfct 11594 prodsnf 11599 fprodm1s 11608 fprodp1s 11609 prodsns 11610 fprodcllemf 11620 fprod2dlemstep 11629 fprodcom2fi 11633 fproddivapf 11638 fprodsplitf 11639 |
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