MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvixpv Structured version   Visualization version   GIF version

Theorem cbvixpv 8166
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 2941 . 2 𝑦𝐵
2 nfcv 2941 . 2 𝑥𝐶
3 cbvixpv.1 . 2 (𝑥 = 𝑦𝐵 = 𝐶)
41, 2, 3cbvixp 8165 1 X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  Xcixp 8148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fn 6104  df-fv 6109  df-ixp 8149
This theorem is referenced by:  funcpropd  16874  invfuc  16948  natpropd  16950  dprdw  18725  dprdwd  18726  ptuni2  21708  ptbasin  21709  ptbasfi  21713  ptpjopn  21744  ptclsg  21747  dfac14  21750  ptcnp  21754  ptcmplem2  22185  ptcmpg  22189  prdsxmslem2  22662  upixp  34012  rrxsnicc  41263  ioorrnopn  41268  ioorrnopnxr  41270  ovnsubadd  41532  hoidmvlelem4  41558  hoidmvle  41560  hspdifhsp  41576  hoiqssbllem2  41583  hspmbl  41589  hoimbl  41591  opnvonmbl  41594  ovnovollem3  41618
  Copyright terms: Public domain W3C validator