Theorem uneq12d 3384

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 3378	. 2 |- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )
4	1, 2, 3	syl2anc 415	1 |- ( ph -> ( A u. C ) = ( B u. D ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   u. cun 3218

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224

This theorem is used by:  disjpr2  3773  diftpsn3  3856  iunxprg  4093  undifexmid  4330  exmidundif  4343  exmidundifim  4344  exmid1stab  4345  suceq  4547  rnpropg  5267  fntpg  5437  fresaunres2disj  5570  foun  5658  fnimapr  5763  fprg  5898  fsnunfv  5916  fsnunres  5917  tfrlemi1  6603  tfr1onlemaccex  6619  tfrcllemaccex  6632  ereq1  6814  mapunen  7151  undifdc  7231  unfiin  7233  djueq12  7379  fztp  10485  fzsuc2  10486  fseq1p1m1  10501  ennnfonelemg  13294  ennnfonelemp1  13297  ennnfonelem1  13298  ennnfonelemnn0  13313  setsvalg  13382  setsfun0  13388  setsresg  13390  setsslid  13403  prdsex  14172  prdsval  14173  psrval  15050  lgsquadlem2  16197  vtxdfifiun  16538  trlsegvdegfi  16708