|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 2904 | . 2 ⊢ Ⅎ𝑘𝐴 | |
| 3 | nfcv 2904 | . 2 ⊢ Ⅎ𝑗𝐴 | |
| 4 | cbvprodi.1 | . 2 ⊢ Ⅎ𝑘𝐵 | |
| 5 | cbvprodi.2 | . 2 ⊢ Ⅎ𝑗𝐶 | |
| 6 | 1, 2, 3, 4, 5 | cbvprod 15950 | 1 ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 Ⅎwnfc 2889 ∏cprod 15940 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-xp 5690 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-iota 6513 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-seq 14044 df-prod 15941 | 
| This theorem is referenced by: prodfc 15982 fprodcllemf 15995 prodsn 15999 prodsnf 16001 fprodm1s 16007 fprodp1s 16008 prodsns 16009 fprod2dlem 16017 fprodcom2 16021 fproddivf 16024 fprodsplitf 16025 | 
| Copyright terms: Public domain | W3C validator |