Theorem uneq12d 3291

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 3285	. 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 1353   u. cun 3128

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134

This theorem is referenced by:  disjpr2  3657  diftpsn3  3734  iunxprg  3968  undifexmid  4194  exmidundif  4207  exmidundifim  4208  exmid1stab  4209  suceq  4403  rnpropg  5109  fntpg  5273  foun  5481  fnimapr  5577  fprg  5700  fsnunfv  5718  fsnunres  5719  tfrlemi1  6333  tfr1onlemaccex  6349  tfrcllemaccex  6362  ereq1  6542  undifdc  6923  unfiin  6925  djueq12  7038  fztp  10078  fzsuc2  10079  fseq1p1m1  10094  ennnfonelemg  12404  ennnfonelemp1  12407  ennnfonelem1  12408  ennnfonelemnn0  12423  setsvalg  12492  setsfun0  12498  setsresg  12500  setsslid  12513  prdsex  12718