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Theorem cbvixpv 8467
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 2974 . 2 𝑦𝐵
2 nfcv 2974 . 2 𝑥𝐶
3 cbvixpv.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbvixp 8466 1 X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  Xcixp 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fn 6351  df-fv 6356  df-ixp 8450
This theorem is referenced by:  funcpropd  17158  invfuc  17232  natpropd  17234  dprdw  19061  dprdwd  19062  ptuni2  22112  ptbasin  22113  ptbasfi  22117  ptpjopn  22148  ptclsg  22151  dfac14  22154  ptcnp  22158  ptcmplem2  22589  ptcmpg  22593  prdsxmslem2  23066  upixp  34885  rrxsnicc  42462  ioorrnopn  42467  ioorrnopnxr  42469  ovnsubadd  42731  hoidmvlelem4  42757  hoidmvle  42759  hspdifhsp  42775  hoiqssbllem2  42782  hspmbl  42788  hoimbl  42790  opnvonmbl  42793  ovnovollem3  42817
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