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Theorem cbvixpv 6850
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 2432 . 2  |-  F/_ y B
2 nfcv 2432 . 2  |-  F/_ x C
3 cbvixpv.1 . 2  |-  ( x  =  y  ->  B  =  C )
41, 2, 3cbvixp 6849 1  |-  X_ x  e.  A  B  =  X_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   X_cixp 6833
This theorem is referenced by:  funcpropd  13790  natpropd  13866  dprdw  15261  ptuni2  17287  ptbasin  17288  ptbasfi  17292  ptpjopn  17322  ptclsg  17325  dfac14  17328  ptcnp  17332  ptcmplem2  17763  ptcmpg  17767  prdsxmslem2  18091  upixp  26506
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fn 5274  df-fv 5279  df-ixp 6834
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