Theorem uneq2 3352

Description: Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
uneq2	|- ( A = B -> ( C u. A ) = ( C u. B ) )

Proof of Theorem uneq2

Step	Hyp	Ref	Expression
1		uneq1 3351	. 2 |- ( A = B -> ( A u. C ) = ( B u. C ) )
2		uncom 3348	. 2 |- ( C u. A ) = ( A u. C )
3		uncom 3348	. 2 |- ( C u. B ) = ( B u. C )
4	1, 2, 3	3eqtr4g 2287	1 |- ( A = B -> ( C u. A ) = ( C u. B ) )

Syntax hints:   -> wi 4   = wceq 1395   u. cun 3195

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201

This theorem is referenced by:  uneq12  3353  uneq2i  3355  uneq2d  3358  uneqin  3455  disjssun  3555  uniprg  3903  sucprc  4503  unexb  4533  unfiexmid  7080  unfidisj  7084  hashunlem  11026  bdunexb  16283  bj-unexg  16284