Theorem uneq12d 3282

Description: Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Hypotheses

Ref	Expression
uneq1d.1	|- ( ph -> A = B )
uneq12d.2	|- ( ph -> C = D )

Assertion

Ref	Expression
uneq12d	|- ( ph -> ( A u. C ) = ( B u. D ) )

Proof of Theorem uneq12d

Step	Hyp	Ref	Expression
1		uneq1d.1	. 2 |- ( ph -> A = B )
2		uneq12d.2	. 2 |- ( ph -> C = D )
3		uneq12 3276	. 2 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )
4	1, 2, 3	syl2anc 409	1 |- ( ph -> ( A u. C ) = ( B u. D ) )

Syntax hints:   -> wi 4   = wceq 1348   u. cun 3119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125

This theorem is referenced by:  disjpr2  3645  diftpsn3  3719  iunxprg  3951  undifexmid  4177  exmidundif  4190  exmidundifim  4191  suceq  4385  rnpropg  5088  fntpg  5252  foun  5459  fnimapr  5554  fprg  5676  fsnunfv  5694  fsnunres  5695  tfrlemi1  6308  tfr1onlemaccex  6324  tfrcllemaccex  6337  ereq1  6516  undifdc  6897  unfiin  6899  djueq12  7012  fztp  10021  fzsuc2  10022  fseq1p1m1  10037  ennnfonelemg  12345  ennnfonelemp1  12348  ennnfonelem1  12349  ennnfonelemnn0  12364  setsvalg  12433  setsfun0  12439  setsresg  12441  setsslid  12453  exmid1stab  13993