Theorem unieqd 3614

Description: Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)

Hypothesis

Ref	Expression
unieqd.1	|- ( ph -> A = B )

Assertion

Ref	Expression
unieqd	|- ( ph -> U. A = U. B )

Proof of Theorem unieqd

Step	Hyp	Ref	Expression
1		unieqd.1	. 2 |- ( ph -> A = B )
2		unieq 3612	. 2 |- ( A = B -> U. A = U. B )
3	1, 2	syl 14	1 |- ( ph -> U. A = U. B )

Syntax hints:   -> wi 4   = wceq 1285   U.cuni 3603

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-uni 3604

This theorem is referenced by:  uniprg  3618  unisng  3620  unisn3  4200  onsucuni2  4309  opswapg  4831  elxp4  4832  elxp5  4833  iotaeq  4899  iotabi  4900  uniabio  4901  funfvdm  5262  funfvdm2  5263  fvun1  5265  fniunfv  5427  funiunfvdm  5428  1stvalg  5794  2ndvalg  5795  fo1st  5809  fo2nd  5810  f1stres  5811  f2ndres  5812  2nd1st  5831  cnvf1olem  5870  brtpos2  5894  dftpos4  5906  tpostpos  5907  recseq  5949  tfrexlem  5977  xpcomco  6360  xpassen  6364  xpdom2  6365  supeq1  6448  supeq2  6451  supeq3  6452  supeq123d  6453  en2other2  6512  dfinfre  8090  sizeinf  9791  sizeennn  9793