Theorem uneq12d 3328

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 3322	. 2 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> ( A u. C ) = ( B u. D ) )

Syntax hints:   -> wi 4   = wceq 1373   u. cun 3164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170

This theorem is referenced by:  disjpr2  3697  diftpsn3  3774  iunxprg  4008  undifexmid  4237  exmidundif  4250  exmidundifim  4251  exmid1stab  4252  suceq  4449  rnpropg  5162  fntpg  5330  foun  5541  fnimapr  5639  fprg  5767  fsnunfv  5785  fsnunres  5786  tfrlemi1  6418  tfr1onlemaccex  6434  tfrcllemaccex  6447  ereq1  6627  undifdc  7021  unfiin  7023  djueq12  7141  fztp  10200  fzsuc2  10201  fseq1p1m1  10216  ennnfonelemg  12774  ennnfonelemp1  12777  ennnfonelem1  12778  ennnfonelemnn0  12793  setsvalg  12862  setsfun0  12868  setsresg  12870  setsslid  12883  prdsex  13101  prdsval  13105  psrval  14428  lgsquadlem2  15555