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  4238  exmidundif  4251  exmidundifim  4252  exmid1stab  4253  suceq  4450  rnpropg  5163  fntpg  5331  foun  5543  fnimapr  5641  fprg  5769  fsnunfv  5787  fsnunres  5788  tfrlemi1  6420  tfr1onlemaccex  6436  tfrcllemaccex  6449  ereq1  6629  undifdc  7023  unfiin  7025  djueq12  7143  fztp  10202  fzsuc2  10203  fseq1p1m1  10218  ennnfonelemg  12807  ennnfonelemp1  12810  ennnfonelem1  12811  ennnfonelemnn0  12826  setsvalg  12895  setsfun0  12901  setsresg  12903  setsslid  12916  prdsex  13134  prdsval  13138  psrval  14461  lgsquadlem2  15588