Theorem uneq12d 3231

Description: Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Hypotheses

Ref	Expression
uneq1d.1	|- ( ph -> A = B )
uneq12d.2	|- ( ph -> C = D )

Assertion

Ref	Expression
uneq12d	|- ( ph -> ( A u. C ) = ( B u. D ) )

Proof of Theorem uneq12d

Step	Hyp	Ref	Expression
1		uneq1d.1	. 2 |- ( ph -> A = B )
2		uneq12d.2	. 2 |- ( ph -> C = D )
3		uneq12 3225	. 2 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )
4	1, 2, 3	syl2anc 408	1 |- ( ph -> ( A u. C ) = ( B u. D ) )

Syntax hints:   -> wi 4   = wceq 1331   u. cun 3069

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075

This theorem is referenced by:  disjpr2  3587  diftpsn3  3661  iunxprg  3893  undifexmid  4117  exmidundif  4129  exmidundifim  4130  suceq  4324  rnpropg  5018  fntpg  5179  foun  5386  fnimapr  5481  fprg  5603  fsnunfv  5621  fsnunres  5622  tfrlemi1  6229  tfr1onlemaccex  6245  tfrcllemaccex  6258  ereq1  6436  undifdc  6812  unfiin  6814  djueq12  6924  fztp  9858  fzsuc2  9859  fseq1p1m1  9874  ennnfonelemg  11916  ennnfonelemp1  11919  ennnfonelem1  11920  ennnfonelemnn0  11935  setsvalg  11989  setsfun0  11995  setsresg  11997  setsslid  12009  exmid1stab  13195