Theorem uneq1d 3288

Description: Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)

Hypothesis

Ref	Expression
uneq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
uneq1d	|- ( ph -> ( A u. C ) = ( B u. C ) )

Proof of Theorem uneq1d

Step	Hyp	Ref	Expression
1		uneq1d.1	. 2 |- ( ph -> A = B )
2		uneq1 3282	. 2 |- ( A = B -> ( A u. C ) = ( B u. C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A u. C ) = ( B u. C ) )

Syntax hints:   -> wi 4   = wceq 1353   u. cun 3127

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 2739  df-un 3133

This theorem is referenced by:  ifeq1  3537  preq1  3669  tpeq1  3678  tpeq2  3679  resasplitss  5392  fmptpr  5705  funresdfunsnss  5716  rdgisucinc  6381  oasuc  6460  omsuc  6468  funresdfunsndc  6502  fisseneq  6926  sbthlemi5  6955  exmidfodomrlemim  7195  fzpred  10063  fseq1p1m1  10087  nn0split  10129  nnsplit  10130  fzo0sn0fzo1  10214  fzosplitprm1  10227  fsum1p  11417  fprod1p  11598  zsupcllemstep  11936  setsvala  12483  setsabsd  12491  setscom  12492