Theorem uneq12i 3269

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 3266	. 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 1342   u. cun 3109

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-un 3115

This theorem is referenced by:  indir  3366  difundir  3370  symdif1  3382  unrab  3388  rabun2  3396  dfif6  3517  dfif3  3528  unopab  4055  xpundi  4654  xpundir  4655  xpun  4659  dmun  4805  resundi  4891  resundir  4892  cnvun  5003  rnun  5006  imaundi  5010  imaundir  5011  dmtpop  5073  coundi  5099  coundir  5100  unidmrn  5130  dfdm2  5132  mptun  5313  fpr  5661  fvsnun2  5677  sbthlemi5  6917  djuunr  7022  djuun  7023  casedm  7042  djudm  7061  djuassen  7164  fz0to3un2pr  10048  fz0to4untppr  10049  fzo0to42pr  10145