Theorem cbviunv 4049

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 2392	. 2 |- F/_ y B
2		nfcv 2392	. 2 |- F/_ x C
3		cbviunv.1	. 2 |- ( x = y -> B = C )
4	1, 2, 3	cbviun 4047	1 |- U_ x e. A B = U_ y e. A C

Syntax hints:   -> wi 4   = wceq 1402   U_ciun 4010

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-iun 4012

This theorem is referenced by:  iunxdif2  4059  ennnfonelemnn0  13294