Theorem uneq12 3353

Description: Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)

Assertion

Ref	Expression
uneq12	|- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )

Proof of Theorem uneq12

Step	Hyp	Ref	Expression
1		uneq1 3351	. 2 |- ( A = B -> ( A u. C ) = ( B u. C ) )
2		uneq2 3352	. 2 |- ( C = D -> ( B u. C ) = ( B u. D ) )
3	1, 2	sylan9eq 2282	1 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )

Syntax hints:   -> wi 4   /\ wa 104   = 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:  uneq12i  3356  uneq12d  3359  un00  3538  opthprc  4770  dmpropg  5201  unixpm  5264  fntpg  5377  fnun  5429  resasplitss  5505  pm54.43  7363