Theorem unieqd 3612

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 3610	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → ∪ 𝐴 = ∪ 𝐵)

Syntax hints:   → wi 4   = wceq 1284  ∪ cuni 3601

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-uni 3602

This theorem is referenced by:  uniprg  3616  unisng  3618  unisn3  4198  onsucuni2  4307  opswapg  4827  elxp4  4828  elxp5  4829  iotaeq  4895  iotabi  4896  uniabio  4897  funfvdm  5257  funfvdm2  5258  fvun1  5260  fniunfv  5422  funiunfvdm  5423  1stvalg  5789  2ndvalg  5790  fo1st  5804  fo2nd  5805  f1stres  5806  f2ndres  5807  2nd1st  5826  cnvf1olem  5865  brtpos2  5889  dftpos4  5901  tpostpos  5902  recseq  5944  tfrexlem  5971  xpcomco  6323  xpassen  6327  xpdom2  6328  supeq1  6399  supeq2  6402  supeq3  6403  supeq123d  6404  dfinfre  8034