Theorem cbvixpv 6610

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

Step	Hyp	Ref	Expression
1		nfcv 2281	. 2 |- F/_ y B
2		nfcv 2281	. 2 |- F/_ x C
3		cbvixpv.1	. 2 |- ( x = y -> B = C )
4	1, 2, 3	cbvixp 6609	1 |- X_ x e. A B = X_ y e. A C

Syntax hints:   -> wi 4   = wceq 1331   X_cixp 6592

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fn 5126  df-fv 5131  df-ixp 6593

This theorem is referenced by: (None)