Theorem uneq12 3574

Description: Equality theorem for the 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 3572	. 2 |- ( A = B -> ( A u. C ) = ( B u. C ) )
2		uneq2 3573	. 2 |- ( C = D -> ( B u. C ) = ( B u. D ) )
3	1, 2	sylan9eq 2525	1 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )

Syntax hints:   -> wi 4   /\ wa 376   = wceq 1452   u. cun 3388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-un 3395

This theorem is referenced by:  uneq12i  3577  uneq12d  3580  un00  3804  opthprc  4887  dmpropg  5316  unixp  5376  fntpg  5644  fnun  5692  resasplit  5765  fvun  5950  rankprb  8340  pm54.43  8452  xpscg  15542  evlseu  18816  ptuncnv  20899  sshjval  27084  bj-2upleq  31676  poimirlem4  32008  poimirlem9  32013  diophun  35687  pwssplit4  36018