Theorem cbvixpv 6564

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 2255	. 2 |- F/_ y B
2		nfcv 2255	. 2 |- F/_ x C
3		cbvixpv.1	. 2 |- ( x = y -> B = C )
4	1, 2, 3	cbvixp 6563	1 |- X_ x e. A B = X_ y e. A C

Syntax hints:   -> wi 4   = wceq 1314   X_cixp 6546

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-iota 5046  df-fn 5084  df-fv 5089  df-ixp 6547

This theorem is referenced by: (None)