Theorem uneq12i 3371

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 3368	. 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 1398   u. cun 3209

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215

This theorem is referenced by:  indir  3470  difundir  3474  symdif1  3486  unrab  3492  rabun2  3500  dfif6  3622  dfif3  3636  unopab  4189  xpundi  4806  xpundir  4807  xpun  4811  dmun  4963  resundi  5051  resundir  5052  cnvun  5168  rnun  5171  imaundi  5175  imaundir  5176  dmtpop  5238  coundi  5264  coundir  5265  unidmrn  5295  dfdm2  5297  mptun  5490  fpr  5866  fvsnun2  5882  sbthlemi5  7231  djuunr  7357  djuun  7358  casedm  7377  djudm  7396  djuassen  7524  fz0to3un2pr  10457  fz0to4untppr  10458  fzo0to42pr  10565  xnn0nnen  10799