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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 2979 | . 2 ⊢ Ⅎ𝑘𝐴 | |
3 | nfcv 2979 | . 2 ⊢ Ⅎ𝑗𝐴 | |
4 | nfcv 2979 | . 2 ⊢ Ⅎ𝑘𝐵 | |
5 | nfcv 2979 | . 2 ⊢ Ⅎ𝑗𝐶 | |
6 | 1, 2, 3, 4, 5 | cbvprod 15271 | 1 ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∏cprod 15261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-iota 6316 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-seq 13373 df-prod 15262 |
This theorem is referenced by: breprexp 31906 mccl 41886 dvnprodlem3 42240 etransclem6 42532 etransclem37 42563 etransclem46 42572 ovnsubadd 42861 hoidmv1le 42883 hoidmvle 42889 hspmbl 42918 ovnovollem3 42947 vonn0ioo 42976 vonn0icc 42977 |
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