Theorem uneq12d 3376

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 3370	. 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 1398   u. cun 3211

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217

This theorem is referenced by:  disjpr2  3755  diftpsn3  3837  iunxprg  4074  undifexmid  4308  exmidundif  4321  exmidundifim  4322  exmid1stab  4323  suceq  4525  rnpropg  5244  fntpg  5414  fresaunres2disj  5547  foun  5635  fnimapr  5739  fprg  5869  fsnunfv  5887  fsnunres  5888  tfrlemi1  6565  tfr1onlemaccex  6581  tfrcllemaccex  6594  ereq1  6776  mapunen  7106  undifdc  7186  unfiin  7188  djueq12  7332  fztp  10416  fzsuc2  10417  fseq1p1m1  10432  ennnfonelemg  13171  ennnfonelemp1  13174  ennnfonelem1  13175  ennnfonelemnn0  13190  setsvalg  13259  setsfun0  13265  setsresg  13267  setsslid  13280  prdsex  13499  prdsval  13503  psrval  14831  lgsquadlem2  15968  vtxdfifiun  16309  trlsegvdegfi  16479