Theorem uneq12i 3274

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 3271	. 2 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )
4	1, 2, 3	mp2an 423	1 |- ( A u. C ) = ( B u. D )

Syntax hints:   = wceq 1343   u. cun 3114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120

This theorem is referenced by:  indir  3371  difundir  3375  symdif1  3387  unrab  3393  rabun2  3401  dfif6  3522  dfif3  3533  unopab  4061  xpundi  4660  xpundir  4661  xpun  4665  dmun  4811  resundi  4897  resundir  4898  cnvun  5009  rnun  5012  imaundi  5016  imaundir  5017  dmtpop  5079  coundi  5105  coundir  5106  unidmrn  5136  dfdm2  5138  mptun  5319  fpr  5667  fvsnun2  5683  sbthlemi5  6926  djuunr  7031  djuun  7032  casedm  7051  djudm  7070  djuassen  7173  fz0to3un2pr  10058  fz0to4untppr  10059  fzo0to42pr  10155