Theorem cbviunv 3847

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 2279	. 2 |- F/_ y B
2		nfcv 2279	. 2 |- F/_ x C
3		cbviunv.1	. 2 |- ( x = y -> B = C )
4	1, 2, 3	cbviun 3845	1 |- U_ x e. A B = U_ y e. A C

Syntax hints:   -> wi 4   = wceq 1331   U_ciun 3808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-iun 3810

This theorem is referenced by:  iunxdif2  3856  ennnfonelemnn0  11924