Theorem cbvixpv 6884

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 2374	. 2 |- F/_ y B
2		nfcv 2374	. 2 |- F/_ x C
3		cbvixpv.1	. 2 |- ( x = y -> B = C )
4	1, 2, 3	cbvixp 6883	1 |- X_ x e. A B = X_ y e. A C

Syntax hints:   -> wi 4   = wceq 1397   X_cixp 6866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fn 5329  df-fv 5334  df-ixp 6867

This theorem is referenced by: (None)