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Mirrors > Home > MPE Home > Th. List > cbvixpv | Structured version Visualization version GIF version |
Description: Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
cbvixpv.1 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbvixpv | ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2764 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2764 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbvixpv.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbvixp 7925 | 1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 Xcixp 7908 |
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-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-iota 5851 df-fn 5891 df-fv 5896 df-ixp 7909 |
This theorem is referenced by: funcpropd 16560 invfuc 16634 natpropd 16636 dprdw 18409 dprdwd 18410 ptuni2 21379 ptbasin 21380 ptbasfi 21384 ptpjopn 21415 ptclsg 21418 dfac14 21421 ptcnp 21425 ptcmplem2 21857 ptcmpg 21861 prdsxmslem2 22334 upixp 33524 rrxsnicc 40520 ioorrnopn 40525 ioorrnopnxr 40527 ovnsubadd 40786 hoidmvlelem4 40812 hoidmvle 40814 hspdifhsp 40830 hoiqssbllem2 40837 hspmbl 40843 hoimbl 40845 opnvonmbl 40848 ovnovollem3 40872 |
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