Theorem cbvixpv 6770

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 2336	. 2 |- F/_ y B
2		nfcv 2336	. 2 |- F/_ x C
3		cbvixpv.1	. 2 |- ( x = y -> B = C )
4	1, 2, 3	cbvixp 6769	1 |- X_ x e. A B = X_ y e. A C

Syntax hints:   -> wi 4   = wceq 1364   X_cixp 6752

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fn 5257  df-fv 5262  df-ixp 6753

This theorem is referenced by: (None)