Theorem uneq1d 3303

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 3297	. 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 1364   u. cun 3142

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148

This theorem is referenced by:  ifeq1  3552  preq1  3684  tpeq1  3693  tpeq2  3694  resasplitss  5410  fmptpr  5724  funresdfunsnss  5735  rdgisucinc  6404  oasuc  6483  omsuc  6491  funresdfunsndc  6525  fisseneq  6949  sbthlemi5  6978  exmidfodomrlemim  7218  fzpred  10088  fseq1p1m1  10112  nn0split  10154  nnsplit  10155  fzo0sn0fzo1  10239  fzosplitprm1  10252  fsum1p  11444  fprod1p  11625  zsupcllemstep  11964  setsvala  12511  setsabsd  12519  setscom  12520  prdsex  12740