Theorem uneq12i 3125

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 3122	. 2 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )
4	1, 2, 3	mp2an 417	1 |- ( A u. C ) = ( B u. D )

Syntax hints:   = wceq 1285   u. cun 2972

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978

This theorem is referenced by:  indir  3214  difundir  3218  symdif1  3230  unrab  3236  rabun2  3244  dfif6  3355  dfif3  3366  unopab  3859  xpundi  4416  xpundir  4417  xpun  4421  dmun  4564  resundi  4647  resundir  4648  cnvun  4753  rnun  4756  imaundi  4760  imaundir  4761  dmtpop  4820  coundi  4846  coundir  4847  unidmrn  4874  dfdm2  4876  mptun  5054  fpr  5371  fvsnun2  5387  fzo0to42pr  9295