Theorem uneq1d 3224

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 3218	. 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 1331   u. cun 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070

This theorem is referenced by:  ifeq1  3472  preq1  3595  tpeq1  3604  tpeq2  3605  resasplitss  5297  fmptpr  5605  funresdfunsnss  5616  rdgisucinc  6275  oasuc  6353  omsuc  6361  funresdfunsndc  6395  fisseneq  6813  sbthlemi5  6842  exmidfodomrlemim  7050  fzpred  9843  fseq1p1m1  9867  nn0split  9906  nnsplit  9907  fzo0sn0fzo1  9991  fzosplitprm1  10004  fsum1p  11180  zsupcllemstep  11627  setsvala  11979  setsabsd  11987  setscom  11988