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| Mirrors > Home > MPE Home > Th. List > cbvprodi | Structured version Visualization version 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 2905 | . 2 ⊢ Ⅎ𝑘𝐴 | |
| 3 | nfcv 2905 | . 2 ⊢ Ⅎ𝑗𝐴 | |
| 4 | cbvprodi.1 | . 2 ⊢ Ⅎ𝑘𝐵 | |
| 5 | cbvprodi.2 | . 2 ⊢ Ⅎ𝑗𝐶 | |
| 6 | 1, 2, 3, 4, 5 | cbvprod 15949 | 1 ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 Ⅎwnfc 2890 ∏cprod 15939 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-seq 14043 df-prod 15940 | 
| This theorem is referenced by: prodfc 15981 fprodcllemf 15994 prodsn 15998 prodsnf 16000 fprodm1s 16006 fprodp1s 16007 prodsns 16008 fprod2dlem 16016 fprodcom2 16020 fproddivf 16023 fprodsplitf 16024 | 
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