Theorem uneq2 3270

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 3269	. 2 |- ( A = B -> ( A u. C ) = ( B u. C ) )
2		uncom 3266	. 2 |- ( C u. A ) = ( A u. C )
3		uncom 3266	. 2 |- ( C u. B ) = ( B u. C )
4	1, 2, 3	3eqtr4g 2224	1 |- ( A = B -> ( C u. A ) = ( C u. B ) )

Syntax hints:   -> wi 4   = wceq 1343   u. cun 3114

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120

This theorem is referenced by:  uneq12  3271  uneq2i  3273  uneq2d  3276  uneqin  3373  disjssun  3472  uniprg  3804  sucprc  4390  unexb  4420  unfiexmid  6883  unfidisj  6887  hashunlem  10717  bdunexb  13802  bj-unexg  13803