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Mirrors > Home > MPE Home > Th. List > cbvprodv | Structured version Visualization version GIF version |
Description: Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.) |
Ref | Expression |
---|---|
cbvprod.1 | ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbvprodv | ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvprod.1 | . 2 ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶) | |
2 | nfcv 2764 | . 2 ⊢ Ⅎ𝑘𝐴 | |
3 | nfcv 2764 | . 2 ⊢ Ⅎ𝑗𝐴 | |
4 | nfcv 2764 | . 2 ⊢ Ⅎ𝑘𝐵 | |
5 | nfcv 2764 | . 2 ⊢ Ⅎ𝑗𝐶 | |
6 | 1, 2, 3, 4, 5 | cbvprod 14645 | 1 ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∏cprod 14635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-iota 5851 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-seq 12802 df-prod 14636 |
This theorem is referenced by: breprexp 30711 mccl 39830 dvnprodlem3 40163 etransclem6 40457 etransclem37 40488 etransclem46 40497 ovnsubadd 40786 hoidmv1le 40808 hoidmvle 40814 hspmbl 40843 ovnovollem3 40872 vonn0ioo 40901 vonn0icc 40902 |
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