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Theorem cbvixpv 8620
Description: Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypothesis
Ref Expression
cbvixpv.1 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbvixpv X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbvixpv
StepHypRef Expression
1 nfcv 2907 . 2 𝑦𝐵
2 nfcv 2907 . 2 𝑥𝐶
3 cbvixpv.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbvixp 8619 1 X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  Xcixp 8602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ral 3069  df-rab 3073  df-v 3425  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4255  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6359  df-fn 6404  df-fv 6409  df-ixp 8603
This theorem is referenced by:  funcpropd  17440  invfuc  17516  natpropd  17518  dprdw  19430  dprdwd  19431  ptuni2  22505  ptbasin  22506  ptbasfi  22510  ptpjopn  22541  ptclsg  22544  dfac14  22547  ptcnp  22551  ptcmplem2  22982  ptcmpg  22986  prdsxmslem2  23459  upixp  35661  rrxsnicc  43562  ioorrnopn  43567  ioorrnopnxr  43569  ovnsubadd  43831  hoidmvlelem4  43857  hoidmvle  43859  hspdifhsp  43875  hoiqssbllem2  43882  hspmbl  43888  hoimbl  43890  opnvonmbl  43893  ovnovollem3  43917
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