Theorem unieqd 3638

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

Hypothesis

Ref	Expression
unieqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
unieqd	⊢ (𝜑 → ∪ 𝐴 = ∪ 𝐵)

Proof of Theorem unieqd

Step	Hyp	Ref	Expression
1		unieqd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		unieq 3636	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → ∪ 𝐴 = ∪ 𝐵)

Syntax hints:   → wi 4   = wceq 1285  ∪ cuni 3627

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 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-uni 3628

This theorem is referenced by:  uniprg  3642  unisng  3644  unisn3  4233  onsucuni2  4342  opswapg  4870  elxp4  4871  elxp5  4872  iotaeq  4941  iotabi  4942  uniabio  4943  funfvdm  5311  funfvdm2  5312  fvun1  5314  fniunfv  5480  funiunfvdm  5481  1stvalg  5847  2ndvalg  5848  fo1st  5862  fo2nd  5863  f1stres  5864  f2ndres  5865  2nd1st  5884  cnvf1olem  5923  brtpos2  5947  dftpos4  5959  tpostpos  5960  recseq  6002  tfrexlem  6030  xpcomco  6471  xpassen  6475  xpdom2  6476  supeq1  6587  supeq2  6590  supeq3  6591  supeq123d  6592  en2other2  6724  dfinfre  8310  hashinfom  10020  hashennn  10022