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Theorem cbvixpv 6618
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 2282 . 2  |-  F/_ y B
2 nfcv 2282 . 2  |-  F/_ x C
3 cbvixpv.1 . 2  |-  ( x  =  y  ->  B  =  C )
41, 2, 3cbvixp 6617 1  |-  X_ x  e.  A  B  =  X_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332   X_cixp 6600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fn 5134  df-fv 5139  df-ixp 6601
This theorem is referenced by: (None)
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