Theorem uneq12d 3384

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 3378	. 2 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )
4	1, 2, 3	syl2anc 415	1 |- ( ph -> ( A u. C ) = ( B u. D ) )

Syntax hints:   -> wi 4   = wceq 1402   u. cun 3218

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224

This theorem is referenced by:  disjpr2  3772  diftpsn3  3854  iunxprg  4091  undifexmid  4328  exmidundif  4341  exmidundifim  4342  exmid1stab  4343  suceq  4545  rnpropg  5265  fntpg  5435  fresaunres2disj  5568  foun  5656  fnimapr  5760  fprg  5892  fsnunfv  5910  fsnunres  5911  tfrlemi1  6596  tfr1onlemaccex  6612  tfrcllemaccex  6625  ereq1  6807  mapunen  7144  undifdc  7224  unfiin  7226  djueq12  7372  fztp  10466  fzsuc2  10467  fseq1p1m1  10482  ennnfonelemg  13275  ennnfonelemp1  13278  ennnfonelem1  13279  ennnfonelemnn0  13294  setsvalg  13363  setsfun0  13369  setsresg  13371  setsslid  13384  prdsex  14152  prdsval  14153  psrval  14976  lgsquadlem2  16114  vtxdfifiun  16455  trlsegvdegfi  16625