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Theorem cbvixpv 7016
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 2523 . 2  |-  F/_ y B
2 nfcv 2523 . 2  |-  F/_ x C
3 cbvixpv.1 . 2  |-  ( x  =  y  ->  B  =  C )
41, 2, 3cbvixp 7015 1  |-  X_ x  e.  A  B  =  X_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   X_cixp 6999
This theorem is referenced by:  funcpropd  14024  invfuc  14098  natpropd  14100  dprdw  15495  ptuni2  17529  ptbasin  17530  ptbasfi  17534  ptpjopn  17565  ptclsg  17568  dfac14  17571  ptcnp  17575  ptcmplem2  18005  ptcmpg  18009  prdsxmslem2  18449  upixp  26122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fn 5397  df-fv 5402  df-ixp 7000
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