Theorem uneq12 3299

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 3297	. 2 |- ( A = B -> ( A u. C ) = ( B u. C ) )
2		uneq2 3298	. 2 |- ( C = D -> ( B u. C ) = ( B u. D ) )
3	1, 2	sylan9eq 2242	1 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   u. cun 3142

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148

This theorem is referenced by:  uneq12i  3302  uneq12d  3305  un00  3484  opthprc  4695  dmpropg  5119  unixpm  5182  fntpg  5291  fnun  5341  resasplitss  5414  pm54.43  7219