Theorem uneq12i 3141

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 3138	. 2 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )
4	1, 2, 3	mp2an 417	1 |- ( A u. C ) = ( B u. D )

Syntax hints:   = wceq 1287   u. cun 2986

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992

This theorem is referenced by:  indir  3237  difundir  3241  symdif1  3253  unrab  3259  rabun2  3267  dfif6  3381  dfif3  3392  unopab  3892  xpundi  4462  xpundir  4463  xpun  4467  dmun  4611  resundi  4694  resundir  4695  cnvun  4803  rnun  4806  imaundi  4810  imaundir  4811  dmtpop  4872  coundi  4898  coundir  4899  unidmrn  4929  dfdm2  4931  mptun  5109  fpr  5442  fvsnun2  5458  sbthlemi5  6614  djuunr  6702  casedm  6721  djudm  6729  fzo0to42pr  9559