Theorem cbvixpv 6826

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 2350	. 2 |- F/_ y B
2		nfcv 2350	. 2 |- F/_ x C
3		cbvixpv.1	. 2 |- ( x = y -> B = C )
4	1, 2, 3	cbvixp 6825	1 |- X_ x e. A B = X_ y e. A C

Syntax hints:   -> wi 4   = wceq 1373   X_cixp 6808

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fn 5293  df-fv 5298  df-ixp 6809

This theorem is referenced by: (None)