Theorem uneq12d 3336

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 3330	. 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 3172

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178

This theorem is referenced by:  disjpr2  3707  diftpsn3  3785  iunxprg  4022  undifexmid  4253  exmidundif  4266  exmidundifim  4267  exmid1stab  4268  suceq  4467  rnpropg  5181  fntpg  5349  foun  5563  fnimapr  5662  fprg  5790  fsnunfv  5808  fsnunres  5809  tfrlemi1  6441  tfr1onlemaccex  6457  tfrcllemaccex  6470  ereq1  6650  undifdc  7047  unfiin  7049  djueq12  7167  fztp  10235  fzsuc2  10236  fseq1p1m1  10251  ennnfonelemg  12889  ennnfonelemp1  12892  ennnfonelem1  12893  ennnfonelemnn0  12908  setsvalg  12977  setsfun0  12983  setsresg  12985  setsslid  12998  prdsex  13216  prdsval  13220  psrval  14543  lgsquadlem2  15670