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

Theorem cbvixpv 6834
Description: Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypothesis
Ref Expression
cbvixpv.1  |-  ( x  =  y  ->  B  =  C )
Assertion
Ref Expression
cbvixpv  |-  X_ x  e.  A  B  =  X_ y  e.  A  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 2419 . 2  |-  F/_ y B
2 nfcv 2419 . 2  |-  F/_ x C
3 cbvixpv.1 . 2  |-  ( x  =  y  ->  B  =  C )
41, 2, 3cbvixp 6833 1  |-  X_ x  e.  A  B  =  X_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   X_cixp 6817
This theorem is referenced by:  funcpropd  13774  natpropd  13850  dprdw  15245  ptuni2  17271  ptbasin  17272  ptbasfi  17276  ptpjopn  17306  ptclsg  17309  dfac14  17312  ptcnp  17316  ptcmplem2  17747  ptcmpg  17751  prdsxmslem2  18075  upixp  26403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fn 5258  df-fv 5263  df-ixp 6818
  Copyright terms: Public domain W3C validator