Theorem uneq12i 3194

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 3191	. 2 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )
4	1, 2, 3	mp2an 420	1 |- ( A u. C ) = ( B u. D )

Syntax hints:   = wceq 1314   u. cun 3035

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041

This theorem is referenced by:  indir  3291  difundir  3295  symdif1  3307  unrab  3313  rabun2  3321  dfif6  3442  dfif3  3453  unopab  3967  xpundi  4555  xpundir  4556  xpun  4560  dmun  4706  resundi  4790  resundir  4791  cnvun  4902  rnun  4905  imaundi  4909  imaundir  4910  dmtpop  4972  coundi  4998  coundir  4999  unidmrn  5029  dfdm2  5031  mptun  5212  fpr  5556  fvsnun2  5572  sbthlemi5  6801  djuunr  6903  djuun  6904  casedm  6923  djudm  6942  djuassen  7021  fzo0to42pr  9890