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Theorem cbvixpv 7072
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 2571 . 2  |-  F/_ y B
2 nfcv 2571 . 2  |-  F/_ x C
3 cbvixpv.1 . 2  |-  ( x  =  y  ->  B  =  C )
41, 2, 3cbvixp 7071 1  |-  X_ x  e.  A  B  =  X_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   X_cixp 7055
This theorem is referenced by:  funcpropd  14089  invfuc  14163  natpropd  14165  dprdw  15560  ptuni2  17600  ptbasin  17601  ptbasfi  17605  ptpjopn  17636  ptclsg  17639  dfac14  17642  ptcnp  17646  ptcmplem2  18076  ptcmpg  18080  prdsxmslem2  18551  upixp  26422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fn 5449  df-fv 5454  df-ixp 7056
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