Theorem uneq12 3370

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 3368	. 2 |- ( A = B -> ( A u. C ) = ( B u. C ) )
2		uneq2 3369	. 2 |- ( C = D -> ( B u. C ) = ( B u. D ) )
3	1, 2	sylan9eq 2287	1 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   u. cun 3211

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217

This theorem is referenced by:  uneq12i  3373  uneq12d  3376  un00  3557  opthprc  4803  dmpropg  5237  unixpm  5300  fntpg  5414  fnun  5466  resasplitss  5546  pm54.43  7489