Theorem uneq12i 3356

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 3353	. 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 1395   u. cun 3195

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201

This theorem is referenced by:  indir  3453  difundir  3457  symdif1  3469  unrab  3475  rabun2  3483  dfif6  3604  dfif3  3616  unopab  4163  xpundi  4775  xpundir  4776  xpun  4780  dmun  4930  resundi  5018  resundir  5019  cnvun  5134  rnun  5137  imaundi  5141  imaundir  5142  dmtpop  5204  coundi  5230  coundir  5231  unidmrn  5261  dfdm2  5263  mptun  5455  fpr  5821  fvsnun2  5837  sbthlemi5  7128  djuunr  7233  djuun  7234  casedm  7253  djudm  7272  djuassen  7399  fz0to3un2pr  10319  fz0to4untppr  10320  fzo0to42pr  10426  xnn0nnen  10659