Theorem uneq12 3456

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 3454	. 2 |- ( A = B -> ( A u. C ) = ( B u. C ) )
2		uneq2 3455	. 2 |- ( C = D -> ( B u. C ) = ( B u. D ) )
3	1, 2	sylan9eq 2456	1 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1649   u. cun 3278

This theorem is referenced by:  uneq12i  3459  uneq12d  3462  un00  3623  opthprc  4884  dmpropg  5302  unixp  5361  fntpg  5465  fnun  5510  resasplit  5572  fvun  5752  rankprb  7733  pm54.43  7843  xpscg  13738  ptuncnv  17792  evlseu  19890  sshjval  22805  diophun  26722  pwssplit4  27059

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-un 3285