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 )

This proof depends on syntax axioms:   = wceq 1402   u. cun 3218

This proof depends on 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 proof 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 used by:  indir  3480  difundir  3484  symdif1  3496  unrab  3504  rabun2  3512  dfif6  3640  dfif3  3654  unopab  4211  xpundi  4833  xpundir  4834  xpun  4838  dmun  4990  resundi  5078  resundir  5079  cnvun  5196  rnun  5199  imaundi  5203  imaundir  5204  dmtpop  5266  coundi  5292  coundir  5293  unidmrn  5323  dfdm2  5325  mptun  5518  fpr  5900  fvsnun2  5916  sbthlemi5  7283  djuunr  7412  djuun  7413  casedm  7432  djudm  7451  djuassen  7579  fz0to3un2pr  10547  fz0to4untppr  10548  fzo0to42pr  10655  xnn0nnen  10895