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Theorem cbvixpv 8909
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 2904 . 2 𝑦𝐵
2 nfcv 2904 . 2 𝑥𝐶
3 cbvixpv.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbvixp 8908 1 X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  Xcixp 8891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fn 6547  df-fv 6552  df-ixp 8892
This theorem is referenced by:  funcpropd  17851  invfuc  17927  natpropd  17929  dprdw  19880  dprdwd  19881  ptuni2  23080  ptbasin  23081  ptbasfi  23085  ptpjopn  23116  ptclsg  23119  dfac14  23122  ptcnp  23126  ptcmplem2  23557  ptcmpg  23561  prdsxmslem2  24038  upixp  36597  rrxsnicc  45016  ioorrnopn  45021  ioorrnopnxr  45023  ovnsubadd  45288  hoidmvlelem4  45314  hoidmvle  45316  hspdifhsp  45332  hoiqssbllem2  45339  hspmbl  45345  hoimbl  45347  opnvonmbl  45350  ovnovollem3  45374
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