Theorem cbviunv 3925

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.)

Hypothesis

Ref	Expression
cbviunv.1	|- ( x = y -> B = C )

Assertion

Ref	Expression
cbviunv	|- U_ x e. A B = U_ y e. A C

Distinct variable groups:   x, A   y, A   y, B   x, C

Allowed substitution hints:   B( x)   C( y)

Proof of Theorem cbviunv

Step	Hyp	Ref	Expression
1		nfcv 2319	. 2 |- F/_ y B
2		nfcv 2319	. 2 |- F/_ x C
3		cbviunv.1	. 2 |- ( x = y -> B = C )
4	1, 2, 3	cbviun 3923	1 |- U_ x e. A B = U_ y e. A C

Syntax hints:   -> wi 4   = wceq 1353   U_ciun 3886

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-iun 3888

This theorem is referenced by:  iunxdif2  3934  ennnfonelemnn0  12414