Theorem uneq12i 3325

Description: Equality inference for union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
uneq1i.1	|- A = B
uneq12i.2	|- C = D

Assertion

Ref	Expression
uneq12i	|- ( A u. C ) = ( B u. D )

Proof of Theorem uneq12i

Step	Hyp	Ref	Expression
1		uneq1i.1	. 2 |- A = B
2		uneq12i.2	. 2 |- C = D
3		uneq12 3322	. 2 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )
4	1, 2, 3	mp2an 426	1 |- ( A u. C ) = ( B u. D )

Syntax hints:   = wceq 1373   u. cun 3164

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170

This theorem is referenced by:  indir  3422  difundir  3426  symdif1  3438  unrab  3444  rabun2  3452  dfif6  3573  dfif3  3584  unopab  4124  xpundi  4732  xpundir  4733  xpun  4737  dmun  4886  resundi  4973  resundir  4974  cnvun  5089  rnun  5092  imaundi  5096  imaundir  5097  dmtpop  5159  coundi  5185  coundir  5186  unidmrn  5216  dfdm2  5218  mptun  5409  fpr  5768  fvsnun2  5784  sbthlemi5  7065  djuunr  7170  djuun  7171  casedm  7190  djudm  7209  djuassen  7331  fz0to3un2pr  10247  fz0to4untppr  10248  fzo0to42pr  10351  xnn0nnen  10584