Theorem uneq12i 3381

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 3378	. 2 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )
4	1, 2, 3	mp2an 430	1 |- ( A u. C ) = ( B u. D )

Syntax hints:   = wceq 1402   u. cun 3218

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224

This theorem is referenced by:  indir  3480  difundir  3484  symdif1  3496  unrab  3504  rabun2  3512  dfif6  3637  dfif3  3651  unopab  4205  xpundi  4826  xpundir  4827  xpun  4831  dmun  4983  resundi  5071  resundir  5072  cnvun  5188  rnun  5191  imaundi  5195  imaundir  5196  dmtpop  5258  coundi  5284  coundir  5285  unidmrn  5315  dfdm2  5317  mptun  5510  fpr  5888  fvsnun2  5904  sbthlemi5  7268  djuunr  7396  djuun  7397  casedm  7416  djudm  7435  djuassen  7563  fz0to3un2pr  10508  fz0to4untppr  10509  fzo0to42pr  10616  xnn0nnen  10852