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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 2903 | . 2 ⊢ Ⅎ𝑘𝐴 | |
3 | nfcv 2903 | . 2 ⊢ Ⅎ𝑗𝐴 | |
4 | cbvprodi.1 | . 2 ⊢ Ⅎ𝑘𝐵 | |
5 | cbvprodi.2 | . 2 ⊢ Ⅎ𝑗𝐶 | |
6 | 1, 2, 3, 4, 5 | cbvprod 15946 | 1 ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Ⅎwnfc 2888 ∏cprod 15936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5695 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-iota 6516 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-seq 14040 df-prod 15937 |
This theorem is referenced by: prodfc 15978 fprodcllemf 15991 prodsn 15995 prodsnf 15997 fprodm1s 16003 fprodp1s 16004 prodsns 16005 fprod2dlem 16013 fprodcom2 16017 fproddivf 16020 fprodsplitf 16021 |
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