Theorem uneq12d 3236

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 3230	. 2 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )
4	1, 2, 3	syl2anc 409	1 |- ( ph -> ( A u. C ) = ( B u. D ) )

Syntax hints:   -> wi 4   = wceq 1332   u. cun 3074

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080

This theorem is referenced by:  disjpr2  3595  diftpsn3  3669  iunxprg  3901  undifexmid  4125  exmidundif  4137  exmidundifim  4138  suceq  4332  rnpropg  5026  fntpg  5187  foun  5394  fnimapr  5489  fprg  5611  fsnunfv  5629  fsnunres  5630  tfrlemi1  6237  tfr1onlemaccex  6253  tfrcllemaccex  6266  ereq1  6444  undifdc  6820  unfiin  6822  djueq12  6932  fztp  9889  fzsuc2  9890  fseq1p1m1  9905  ennnfonelemg  11952  ennnfonelemp1  11955  ennnfonelem1  11956  ennnfonelemnn0  11971  setsvalg  12028  setsfun0  12034  setsresg  12036  setsslid  12048  exmid1stab  13368