Theorem uneq12d 3302

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 3296	. 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 1363   u. cun 3139

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145

This theorem is referenced by:  disjpr2  3668  diftpsn3  3745  iunxprg  3979  undifexmid  4205  exmidundif  4218  exmidundifim  4219  exmid1stab  4220  suceq  4414  rnpropg  5120  fntpg  5284  foun  5492  fnimapr  5589  fprg  5712  fsnunfv  5730  fsnunres  5731  tfrlemi1  6347  tfr1onlemaccex  6363  tfrcllemaccex  6376  ereq1  6556  undifdc  6937  unfiin  6939  djueq12  7052  fztp  10092  fzsuc2  10093  fseq1p1m1  10108  ennnfonelemg  12418  ennnfonelemp1  12421  ennnfonelem1  12422  ennnfonelemnn0  12437  setsvalg  12506  setsfun0  12512  setsresg  12514  setsslid  12527  prdsex  12736