Theorem uneq12i 3311

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 3308	. 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 1364   u. cun 3151

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 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157

This theorem is referenced by:  indir  3408  difundir  3412  symdif1  3424  unrab  3430  rabun2  3438  dfif6  3559  dfif3  3570  unopab  4108  xpundi  4715  xpundir  4716  xpun  4720  dmun  4869  resundi  4955  resundir  4956  cnvun  5071  rnun  5074  imaundi  5078  imaundir  5079  dmtpop  5141  coundi  5167  coundir  5168  unidmrn  5198  dfdm2  5200  mptun  5385  fpr  5740  fvsnun2  5756  sbthlemi5  7020  djuunr  7125  djuun  7126  casedm  7145  djudm  7164  djuassen  7277  fz0to3un2pr  10189  fz0to4untppr  10190  fzo0to42pr  10287  xnn0nnen  10508