Theorem uneq12i 3302

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 3299	. 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 1364   u. cun 3142

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148

This theorem is referenced by:  indir  3399  difundir  3403  symdif1  3415  unrab  3421  rabun2  3429  dfif6  3551  dfif3  3562  unopab  4097  xpundi  4700  xpundir  4701  xpun  4705  dmun  4852  resundi  4938  resundir  4939  cnvun  5052  rnun  5055  imaundi  5059  imaundir  5060  dmtpop  5122  coundi  5148  coundir  5149  unidmrn  5179  dfdm2  5181  mptun  5366  fpr  5719  fvsnun2  5735  sbthlemi5  6990  djuunr  7095  djuun  7096  casedm  7115  djudm  7134  djuassen  7246  fz0to3un2pr  10153  fz0to4untppr  10154  fzo0to42pr  10250