Theorem uneq12i 3289

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 3286	. 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 1353   u. cun 3129

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135

This theorem is referenced by:  indir  3386  difundir  3390  symdif1  3402  unrab  3408  rabun2  3416  dfif6  3538  dfif3  3549  unopab  4084  xpundi  4684  xpundir  4685  xpun  4689  dmun  4836  resundi  4922  resundir  4923  cnvun  5036  rnun  5039  imaundi  5043  imaundir  5044  dmtpop  5106  coundi  5132  coundir  5133  unidmrn  5163  dfdm2  5165  mptun  5349  fpr  5700  fvsnun2  5716  sbthlemi5  6962  djuunr  7067  djuun  7068  casedm  7087  djudm  7106  djuassen  7218  fz0to3un2pr  10125  fz0to4untppr  10126  fzo0to42pr  10222