Theorem uneq12d 3360

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 3354	. 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 1395   u. cun 3196

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202

This theorem is referenced by:  disjpr2  3731  diftpsn3  3812  iunxprg  4049  undifexmid  4281  exmidundif  4294  exmidundifim  4295  exmid1stab  4296  suceq  4497  rnpropg  5214  fntpg  5383  foun  5599  fnimapr  5702  fprg  5832  fsnunfv  5850  fsnunres  5851  tfrlemi1  6493  tfr1onlemaccex  6509  tfrcllemaccex  6522  ereq1  6704  undifdc  7109  unfiin  7111  djueq12  7229  fztp  10303  fzsuc2  10304  fseq1p1m1  10319  ennnfonelemg  13014  ennnfonelemp1  13017  ennnfonelem1  13018  ennnfonelemnn0  13033  setsvalg  13102  setsfun0  13108  setsresg  13110  setsslid  13123  prdsex  13342  prdsval  13346  psrval  14670  lgsquadlem2  15797  vtxdfifiun  16103