Theorem uneq12d 3362

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 3356	. 2 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> ( A u. C ) = ( B u. D ) )

Syntax hints:   -> wi 4   = wceq 1397   u. cun 3198

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204

This theorem is referenced by:  disjpr2  3733  diftpsn3  3814  iunxprg  4051  undifexmid  4283  exmidundif  4296  exmidundifim  4297  exmid1stab  4298  suceq  4499  rnpropg  5216  fntpg  5386  foun  5603  fnimapr  5707  fprg  5838  fsnunfv  5856  fsnunres  5857  tfrlemi1  6501  tfr1onlemaccex  6517  tfrcllemaccex  6530  ereq1  6712  undifdc  7119  unfiin  7121  djueq12  7241  fztp  10316  fzsuc2  10317  fseq1p1m1  10332  ennnfonelemg  13045  ennnfonelemp1  13048  ennnfonelem1  13049  ennnfonelemnn0  13064  setsvalg  13133  setsfun0  13139  setsresg  13141  setsslid  13154  prdsex  13373  prdsval  13377  psrval  14702  lgsquadlem2  15834  vtxdfifiun  16175  trlsegvdegfi  16345